This package contains functions for computing the decomposition matrices for Iwahori--Hecke algebras of the symmetric groups. As the (modular) representation theory of these algebras closely resembles that of the (modular) representation theory of the symmetric groups --- indeed, the later is a special case of the former --- many of the combinatorial tools from the representation theory of the symmetric group are included in the package.
These programs grew out of the attempts by Gordon James and myself [JM1] to understand the decomposition matrices of Hecke algebras of type A when <q>=-1. The package is now much more general and its highlights include:
1. Specht provides a means of working in the Grothendieck ring of a Hecke algebra H using the three natural bases corresponding to the Specht modules, projective indecomposable modules, and simple modules.
2. For Hecke algebras defined over fields of characteristic zero we have implemented the algorithm of Lascoux, Leclerc, and Thibon [LLT] for computing decomposition numbers and ``crystallized decomposition matrices''. In principle, this gives all of the decomposition matrices of Hecke algebras defined over fields of characteristic zero.
3. We provide a way of inducing and restricting modules. In addition, it is possible to ``induce'' decomposition matrices; this function is quite effective in calculating the decomposition matrices of Hecke algebras for small n.
4. The q--analogue of Schaper's theorem [JM2] is included, as is Kleshchev's [K] algorithm of calculating the Mullineux map. Both are used extensively when inducing decomposition matrices.
5. Specht can be used to compute the decomposition numbers of q--Schur algebras (and the general linear groups), although there is less direct support for these algebras. The decomposition matrices for the q--Schur algebras defined over fields of characteristic zero for n<11 and all e are included in Specht.
6. The Littlewood--Richard rule, its inverse, and functions for many of the standard operations on partitions (such as calculating cores, quotients, and adding and removing hooks), are included.
7. The decomposition matrices for the symmetric groups Sym_n are included for n<15 and for all primes.
The modular representation theory of Hecke algebras
The ``modular'' representation theory of the Iwahori--Hecke algebras of type A was pioneered by Dipper and James [DJ1,DJ2]; here we briefly outline the theory, referring the reader to the references for details. Iwahori-Hecke algebras; see also Hecke.
Given a commutative integral domain R and a non--zero unit q in R,
let <H>=<H>_{<R>, <q>} be the Hecke algebra of the symmetric group
Sym_n on n symbols defined over R and with parameter q. For
each partition mu of n, Dipper and James defined a Specht
module S(mu). Let rad S(mu) be the radical of S(mu)
and define D(mu)=S(mu)/rad S(mu). When R is a field,
D(mu) is either zero or absolutely irreducible. Henceforth, we will
always assume that R is a field.
Given a non--negative integer i, let [i]_q=1+q+ldots+q^{i-1}. Define e to be the smallest non--negative integer such that [<e>]_q=0; if no such integer exists, we set e equal to 0. Many of the functions in this package depend upon e; the integer e is the Hecke algebras analogue of the characteristic of the field in the modular representation theory of finite groups.
A partition mu=(mu_1,mu_2,ldots) is e--singular if there exists
an integer i such that mu_i=mu_{i+1}=cdots=mu_{i+<e>-1}>0;
otherwise, mu is e--regular. Dipper and James [DJ1] showed that
D(nu)ne(0) if and only if nu is e--regular and that the
D(nu) give a complete set of non--isomorphic irreducible
H--modules as nu runs over the e--regular partitions of n.
Further, S(mu) and S(nu) belong to the same block if and
only if mu and nu have the same e-core [DJ2,JM2]. Note that
these results depend only on e and not directly on R or q.
Given two partitions mu and nu, where nu is e--regular, let
d_{munu} be the composition multiplicity of D(nu) in
S(mu). The matrix D=(d_{munu}) is the decomposition matrix
of H. When the rows and columns are ordered in a way compatible with
dominance, D is lower unitriangular.
The indecomposable H-modules P(nu) are indexed by e-regular
partitions nu. By general arguments, P(nu) has the same
composition factors as
sum_{mu} d_{munu} 'S'(<mu>) ;
so these linear combinations of modules become identified in the
Grothendieck ring of H. Similarly,
'D'(<nu>) = sum_{mu} d_{numu}^{-1} 'S'(<mu>)
in the Grothendieck ring. These observations are the basis for many of
the computations in Specht.
Two small examples
Because of the algorithm of [LLT], in principle, all of decomposition matrices for all Hecke algebras defined over fields of characteristic zero are known and available using Specht. The algorithm is recursive; however, it is quite quick and, as with a car, you need never look at the engine:
gap> H:=Specht(4); # e=4, 'R' a field of characteristic 0 Specht(e=4, S(), P(), D(), Pq()) gap> InducedModule(H.P(12,2)); P(13,2)+P(12,3)+P(12,2,1)+P(10,3,2)+P(9,6)
The [LLT] algorithm was applied 24 times during this calculation.
For Hecke algebras defined over fields of positive characteristic the major tool provided by Specht, apart from the decomposition matrices contained in the libraries, is a way of ``inducing'' decomposition matrices. This makes it fairly easy to calculate the associated decomposition matrices for ``small'' n. For example, the Specht libraries contain the decomposition matrices for the symmetric groups Sym_n over fields of characteristic 3 for n<15. These matrices were calculated by Specht using the following commands:
gap> H:=Specht(3,3); # e=3, 'R' field of characteristic 3 Specht(e=3, p=3, S(), P(), D()) gap> d:=DecompositionMatrix(H,5); # known for $n\<2e$ 5 | 1 4,1 | . 1 3,2 | . 1 1 3,1^2 | . . . 1 2^2,1 | 1 . . . 1 2,1^3 | . . . . 1 1^5 | . . 1 . . gap> for n in [6..14] do > d:=InducedDecompositionMatrix(d); SaveDecompositionMatrix(d); > od;
The function InducedDecompositionMatrix contains almost every trick
that I know for computing decomposition matrices (except using the spin
groups). I would be very happy to hear of any improvements.
Specht can also be used to calculate the decomposition numbers of the q--Schur algebras; although, as yet, here no additional routines for calculating the projective indecomposables indexed by e--singular partitions. Such routines will probably be included in a future release, together with the (conjectural) algorithm [LT] for computing the decomposition matrices of the q--Schur algebras over fields of characteristic zero.
In the next release of Specht, I will also include functions for
computing the decomposition matrices of Hecke algebras of type B, and
more generally those of the Ariki--Koike algebras. As with the Hecke
algebra of type A, there is an algorithm for computing the decomposition
matrices of these algebras when R is a field of characteristic zero [M].
Credits
I would like to thank Gordon James, Johannes Lipp, and Klaus Lux for their comments and suggestions.
If you find Specht useful please let me know. I would also appreciate hearing any suggestions, comments, or improvements. In addition, if Specht does play a significant role in your research, please send me a copy of the paper(s) and please cite Specht in your references.
The lastest version of Specht can be obtained from http://www.maths.usyd.edu.au:8000/u/mathas/specht/.
Andrew Mathas
mathas@maths.usyd.edu.au
University of Sydney, 1997.
Supported in part by SERC grant GR/J37690
References
[A] S. Ariki,
On the decomposition numbers of the Hecke algebra of G(m,1,n),
J. Math. Kyoto Univ., 36 (1996), 789--808.
[B] J. Brundan,
Modular branching rules for quantum GL_n and the Hecke algebra
of type A, Proc. London Math. Soc (3), to appear.
[DJ1] R. Dipper and G. James,
Representations of Hecke algebras of general linear groups,
Proc. London Math. Soc. (3), 52 (1986), 20--52.
[DJ2] R. Dipper and G. James,
Blocks and idempotents of Hecke algebras of general linear groups,
Proc. London Math. Soc. (3), 54 (1987), 57--82.
[G] M. Geck,
Brauer trees of Hecke algebras, Comm. Alg., 20 (1992), 2937--2973.
[Gr] I. Grojnowski,
Affine Hecke algebras (and affine quantum GL_n) at roots of unity,
IMRN~5 (1994), 215--217.
[J] G. James,
The decomposition matrices of GL_n(q) for n le 10,
Proc. London Math. Soc.,~60 (1990), 225--264.
[JK] G. James and A. Kerber,
The representation theory of the symmetric group, 16,
Encyclopedia of Mathematics, Addison--Wesley, Massachusetts~(1981).
[JM1] G. James and A. Mathas,
Hecke algebras of type A at q=-1, J. Algebra,
184 (1996), 102--158.
[JM2] G. James and A. Mathas,
A q--analogue of the Jantzen--Schaper Theorem, Proc. London Math.
Soc. (3), 74, 1997, 241--274.
[K] A. Kleshchev,
Branching rules for modular representations III,
J. London Math. Soc., 54, 1996, 25--38.
[LLT] A. Lascoux, B. Leclerc, and J-Y. Thibon,
Hecke algebras at roots of unity and crystal bases of quantum
affine algebras, Comm. Math. Phys., 181 (1996), 205--263.
[LT] B. Leclerc and J-Y. Thibon,
Canonical bases and q--deformed Fock spaces, Int. Research Notices
9 (1996), 447--456.
[M] A. Mathas,
Canonical bases and the decomposition matrices of Ariki--Koike
algebras, preprint~1996.
Subsections
Specht(e)
Specht(e, p)
Specht(e, p, val [,HeckeRing])
Let R be a field of characteristic 0, q a non--zero element of R,
and let e be the smallest positive integer such that
1+q+ldots+q^e-1=0
(we set <e>=0 if no such integer exists). The record returned
by Specht(e) allows calculations in the Grothendieck rings of
the Hecke algebras H of type A which are defined over R and
have parameter q. (The Hecke algebra is described in Chapter
Iwahori-Hecke algebras; see also Hecke Hecke.) Below we also
describe how to consider Hecke algebras defined over fields of positive
characteristic.
Specht returns a record which contains, among other things, functions
S , P, and D which correspond to the Specht modules, projective
indecomposable modules, and the simple modules for the family of Hecke
algebras determined by R and q. Specht allows manipulation of
arbitrary linear combinations of these ``modules'', as well as a way
of inducing and restricting them, ``multiplying'' them, and converting
between these three natural bases of the Grothendieck ring. Multiplication
of modules corresponds to taking a tensor product, and then inducing (thus
giving a module for a larger Hecke algebra).
gap> RequirePackage("specht"); H:=Specht(5);
Specht(e=5, S(), P(), D(), Pq())
gap> H.D(3,2,1);
D(3,2,1)
gap> H.S( last );
S(6)-S(4,2)+S(3,2,1)
gap> InducedModule(H.P(3,2,1));
P(4,2,1)+P(3,3,1)+P(3,2,2)+2*P(3,2,1,1)
gap> H.S(last);
S(4,2,1)+S(3,3,1)+S(3,2,2)+2*S(3,2,1,1)+S(2,2,2,1)+S(2,2,1,1,1)
gap> H.D(3,1)*H.D(3);
D(7)+2*D(6,1)+D(5,2)+D(5,1,1)+2*D(4,3)+D(4,2,1)+D(3,3,1)
gap> RestrictedModule(last);
4*D(6)+3*D(5,1)+5*D(4,2)+2*D(4,1,1)+2*D(3,3)+2*D(3,2,1)
gap> H.S(last);
S(6)+3*S(5,1)+3*S(4,2)+2*S(4,1,1)+2*S(3,3)+2*S(3,2,1)
gap> H.P(last);
P(6)+3*P(5,1)+2*P(4,2)+2*P(4,1,1)+2*P(3,3)
The way in which the partitions indexing the modules are printed can
be changed using SpechtPrettyPrint SpechtPrettyPrint.
There is also a function Schur Schur for doing calculations with
the q--Schur algebra. See DecompositionMatrix DecompositionMatrix,
and CrystalDecompositionMatrix CrystalDecompositionMatrix.
This function requires the package ``specht'' (see RequirePackage).
The functions H.S, H.P, and H.D
The functions H.S, H.P, and H.D return records which correspond
to Specht modules, projective indecomposable modules, and simple
modules respectively. Each of these three functions can be called in
four different ways, as we now describe.
H.S(mu) H.P(mu) H.D(mu)
In the first form, mu is a partition (either a list, or a sequence of integers), and the corresponding Specht module, PIM, or simple module (respectively), is returned.
gap> H.P(4,3,2); P(4,3,2)
H.S(x) H.P(x) H.D(x)
Here, x is an H--module. In this form, H.S rewrites x as a linear
combination of Specht modules, if possible. Similarly, H.P and H.D
rewrite x as a linear combination of PIMs and simple modules
respectively. These conversions require knowledge of the relevant
decomposition matrix of H; if this is not known then false is
returned (over fields of characteristic zero, all of the decomposition
matrices are known via the algorithm of [LLT]; various other
decomposition matrices are included with Specht). For example,
H.S(H.P(mu)) returns
sum_nu d_numu S(nu),
or false if some of these decomposition multiplicities are not known.
gap> H.D( H.P(4,3,2) ); D(5,3,1)+2*D(4,3,2)+D(2,2,2,2,1) gap> H.S( H.D( H.S(1,1,1,1,1) ) ); -S(5)+S(4,1)-S(3,1,1)+S(2,1,1,1)
As the last example shows, Specht does not always behave as expected.
The reason for this is that Specht modules indexed by e--singular
partitions can always be written as a linear combination of Specht
modules which involve only e--regular partitions. As such, it is not
always clear when two elements are equal in the Grothendieck ring.
Consequently, to test whether two modules are equal you should first
rewrite both modules in the D--basis; this is not done by Specht
because it would be very inefficient.
H.S(d, mu) H.P(d, mu) H.D(d, mu)
In the third form, d is a decomposition matrix and mu is a
partition. This is useful when you are trying to calculate a new
decomposition matrix d because it allows you to do calculations using
the known entries of d to deduce information about the unknown ones.
When used in this way, H.P and H.D use d to rewrite P(mu) and
D(mu) respectively as a linear combination of Specht modules, and
H.S uses d to write S(mu) as a linear combination of simple
modules. If the values of the unknown entries in d are needed, false
is returned.
gap> H:=Specht(3,3); # e = 3, p = 3 = characteristic of 'R'
Specht(e=3, p=3, S(), P(), D())
gap> d:=InducedDecompositionMatrix(DecompositionMatrix(H,14));;
# Inducing....
The following projectives are missing from <d>:
[ 15 ] [ 8, 7 ]
gap> H.P(d,4,3,3,2,2,1);
S(4,3,3,2,2,1)+S(4,3,3,2,1,1,1)+S(4,3,2,2,2,1,1)+S(3,3,3,2,2,1,1)
gap> H.S(d,7, 3, 3, 2);
D(11,2,1,1)+D(10,3,1,1)+D(8,5,1,1)+D(8,3,3,1)+D(7,6,1,1)+D(7,3,3,2)
gap> H.D(d,14,1);
false
The final example returned false because the partitions (14,1)
and (15) have the same 3--core (and P(15) is missing from d).
H.S(d, x) H.P(d, x) H.D(d, x)
In the final form, d is a decomposition matrix and x is a module. All
three functions rewrite x in their respective basis using d. Again
this is only useful when you are trying to calculate a new decomposition
matrix because, for any ``known'' decomposition matrix d, H.S(x)
and H.S(d, x) are equivalent (and similarly for H.P and H.D).
gap> H.S(d, H.D(d,10,5) ); -S(13,2)+S(10,5)
Specht(p, p) can be used to consider the group algebras of
the symmetric groups over fields of characteristic p (i.e. e=p, and
R a field of characteristic~p).
For example, the dimensions of the simple modules of Sym_6 over fields of characteristic 5 can be computed as follows:
gap> H:=Specht(5,5);; SimpleDimension(H,6); 6 : 1 5,1 : 5 4,2 : 8 4,1^2 : 10 3^2 : 5 3,2,1 : 8 3,1^3 : 10 2^3 : 5 2^2,1^2 : 1 2,1^4 : 5
Specht must
also be supplied with a valuation map val as an argument. The
function val is a map from some PID into the natural numbers; at
present it is needed only by functions which rely (at least implicitly),
upon the q--analogue of Schaper's theorem. In general, val depends
upon q and the characteristic of R; full details can be found in [JM2].
Over fields of characteristic zero, and in the symmetric group case, the
function val is automatically defined by Specht. When R is a field
of characteristic zero, val([i]_q) is 1 if e divides i and~0
otherwise (this is the valuation map associated to the prime ideal in
C[v] generated by the e--th cyclotomic polynomial). When <q>=1
and R is a field of characteristic p, val is the usual p--adic
valuation map.
As another example, if <q>=4 and R is a field of characteristic 5 (so <e>=2), then the valuation map sends the integer x to nu_5([4]_x) where [4]_x is interpreted as an integer and nu_5 is the usual 5--adic valuation. To consider this Hecke algebra one could proceed as follows:
gap> val:=function(x) local v;
> x:=Sum([0..x-1],v->4^v); # x-${>}$[x]\_q
> v:=0; while x mod 5=0 do x:=x/5; v:=v+1; od;
> return v;
> end;;
gap> H:=Specht(2,5,val,"e2q4");
Specht(e=2, p=5, S(), P(), D(), HeckeRing="e2q4")
Notice the string ``e2q4'' which was also passed to Specht in this
example. Although it is not strictly necessary, it is a good idea when
using a ``non--standard'' valuation map val to specify the value
of H.HeckeRing=HeckeRing. This string is used for internal
bookkeeping by Specht; in particular, it is used to determine filenames
when reading and saving decomposition matrices. If a ``standard''
valuation map is used then HeckeRing is set to the string
``e{<}e{>}p{<}p{>}''; otherwise it defaults to ``unknown''. The
function SaveDecompositionMatrix will not save any decomposition
matrix for any Hecke algebra H with H.HeckeRing=``unknown''.
The Fock space F is an (integrable) module for the quantum group
U_q(widehat{sl}_{<e>}) of the affine special linear group. F is a
free C[v]--module with basis the set of all Specht modules
S(mu) for all partitions mu of all integers
F = bigoplus_nge0bigoplus_muvdash nC[v] S(mu);
here v=H.info.Indeterminate is an indeterminate over the integers
(or strictly, C). The canonical basis elements Pq(mu) for the
U_q(widehat{sl}_e)--submodule of F generated by the 0--partition
are indexed by e--regular partitions mu. Moreover, under
specialization, Pq(mu) maps to P(mu). An eloquent
description of the algorithm for computing H.Pq(mu) can be found
in [LLT].
To access the elements of the Fock space Specht provides the functions:
H.Pq(mu) H.Sq(mu)
Notice that, unlike H.P and H.S the only arguments which H.Pq and
H.Sq accept are partitions. (Given that our indeterminate is v these
functions should really be called H.Pv and H.Sv; here ``q'' stands
for ``quantum.)
The function H.Pq computes the canonical basis element Pq(mu)
of the Fock space corresponding to the e--regular partition mu
(there is a canonical basis --- defined using a larger quantum group ---
for the whole of the Fock space [LT]; conjecturally, this basis can be used
to compute the decomposition matrices for the q--Schur algebra over fields
of characteristic zero). The second function returns a standard basis
element S(mu) of F.
gap> H:=Specht(4); Specht(e=4, S(), P(), D(), Pq()) gap> H.Pq(6,2); S(6,2)+v*S(5,3) gap> RestrictedModule(last); S(6,1)+(v + v^(-1))*S(5,2)+v*S(4,3) gap> H.P(last); P(6,1)+(v + v^(-1))*P(5,2) gap> Specialized(last); P(6,1)+2*P(5,2) gap> H.Sq(5,3,2); S(5,3,2) gap> InducedModule(last,0); v^(-1)*S(5,3,3)
The modules returned by H.Pq and H.Sq behave very much like elements
of the Grothendieck ring of H; however, they should be considered as
elements of the Fock space. The key difference is that when induced or
restricted ``quantum'' analogues of induction and restriction are used.
These analogues correspond to the action of U_q(widehat{sl}_{<e>})
on F[LLT].
In effect, the functions H.Pq and H.Sq allow computations in
the Fock space, using the functions InducedModule InducedModule and
RestrictedModule RestrictedModule. The functions H.S, H.P, and
H.D can also be applied to elements of the Fock space, in which case
they have the expected effect. In addition, any element of the Fock space
can be specialized to give the corresponding element of the Grothendieck
ring of H (it is because of this correspondence that we do not make a
distinction between elements of the Fock space and the Grothendieck
ring of H).
When working over fields of characteristic zero Specht will
automatically calculate any canonical basis elements that it needs for
computations in the Grothendieck ring of H. If you are not interested
in the canonical basis elements you need never work with them directly.
If, for some reason, you do not want Specht to use the canonical basis
elements to calculate decomposition numbers then all you need to do is
Unbind(H.Pq).
Schur(e)
Schur(e, p)
Schur(e, p, val [,HeckeRing])
This function behaves almost identically to the function Specht (see
Specht), the only difference being that the three functions in the
record S returned by Schur are called S.W, S.P, and S.F
and that they correspond to the q-Weyl modules, the projective
decomposable modules, and the simple modules of the q--Schur algebra
respectively. Note that our labeling of these modules is non--standard,
following that used by James in [J]. The standard labeling can be
obtained from ours by replacing all partitions by their conjugates.
Almost all of the functions in Specht which accept a Specht
record H will also accept a record S returned by Schur
In the current version of Specht the decomposition matrices of q--Schur
algebras are not fully supported. The InducedDecompositionMatrix
function can be applied to these matrices; however there are no additional
routines available for calculating the columns corresponding to
e--singular partitions. The decomposition matrices for the q--Schur
algebras defined over a field of characteristic 0 for <n> le 10 are in
the Specht libraries.
gap> S:=Schur(2); Schur(e=2, W(), P(), F(), Pq()) gap> InducedDecompositionMatrix(DecompositionMatrix(S,3)); # The following projectives are missing from <d>: # [ 2, 2 ] 4 | 1 # 'DecompositionMatrix'(S,4) returns the 3,1 | 1 1 # full decomposition matrix. The point 2^2 | . 1 . # of this example is to emphasize the 2,1^2 | 1 1 . 1 # current limitations of 'Schur'. 1^4 | 1 . . 1 1
Note that when S is defined over a field of characteristic zero then
it contains a function S.Pq for calculating canonical basis elements
(see Specht Specht); currently S.Pq(mu) is implemented only
for e--regular partitions. There is also a function H.Wq.
See also Specht Specht. This function requires the package
``specht'' (see RequirePackage).
DecompositionMatrix(H, n [,Ordering])
DecompositionMatrix(H, filename [,Ordering])
The function DecompositionMatrix returns the decomposition matrix D of
'H'(Sym_n) where H is a Hecke algebra record returned by the function
Specht (or Schur). DecompositionMatrix first checks to see whether
the required decomposition matrix exists as a library file (checking first
in the current directory, next in the directory specified by
SpechtDirectory, and finally in the Specht libraries). If H.Pq
exists, DecompositionMatrix next looks for crystallized decomposition
matrices (see CrystalDecompositionMatrix CrystalDecompositionMatrix).
If the decomposition matrix d is not stored in the library
DecompositionMatrix will calculate d when H is a Hecke algebra with a
base field R of characteristic zero, and will return false otherwise
(in which case the function CalculateDecompositionMatrix
CalculateDecompositionMatrix can be used to force Specht to try and
calculate this matrix).
For Hecke algebras defined over fields of characteristic zero, Specht
uses the algorithm of [LLT] to calculate decomposition matrices
(this feature can be disabled by unbinding H.Pq). The decomposition
matrices for the q--Schur algebras for <n> le 10 are contained in the
Specht library, as are those for the symmetric group over fields of
positive characteristic when <n><15.
Once a decomposition matrix is known, Specht keeps an internal copy
of it which is used by the functions H.S, H.P, and H.D; these
functions also read decomposition matrix files as needed.
If you set the variable SpechtDirectory, then Specht will also search
for decomposition matrix files in this directory. The files in the current
directory override those in SpechtDirectory and those in the Specht
libraries.
In the second form of the function, when a filename is supplied,
DecompositionMatrix will read the decomposition matrix in the file
filename, and this matrix will become Specht's internal copy of
this matrix.
By default, the rows and columns of the decomposition matrices are ordered
lexicographically. This can be changed by supplying DecompositionMatrix
with an ordering function such as LengthLexicographic or
ReverseDominance. You do not need to specify the ordering you want
every time you call DecompositionMatrix; Specht will keep the same
ordering until you change it again. This ordering can also be set ``by
hand'' using the variable H.Ordering. indexH.Ordering!Specht
gap> DecompositionMatrix(Specht(3),6,LengthLexicographic); 6 | 1 5,1 | 1 1 4,2 | . . 1 3^2 | . 1 . 1 4,1^2 | . 1 . . 1 3,2,1 | 1 1 . 1 1 1 2^3 | 1 . . . . 1 3,1^3 | . . . . 1 1 2^2,1^2| . . . . . . 1 2,1^4 | . . . 1 . 1 . 1^6 | . . . 1 . . .Once you have a decomposition matrix it is often nice to be able to print it. The on screen version is often good enough; there is also a
TeXcommand which generates a LaTeX version. There are also functions for converting Specht decomposition matrices into GAP matrices and visa versa (seeMatrixDecompositionMatrixMatrixDecompositionMatrix andDecompositionMatrixMatrixDecompositionMatrixMatrix).Using the function
InducedDecompositionMatrix(see InducedDecompositionMatrix), it is possible to induce a decomposition matrix. See alsoSaveDecompositionMatrixSaveDecompositionMatrix andIsNewIndecomposableIsNewIndecomposable,SpechtSpecht,SchurSchur, andCrystalDecompositionMatrixCrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
71.4 CrystalDecompositionMatrix
CrystalDecompositionMatrix(H, n [,Ordering])
CrystalDecompositionMatrix(H, filename [,Ordering])This function is similar to
DecompositionMatrix, except that it returns a crystallized decomposition matrix. The columns of decomposition matrices correspond to projective indecomposables; the columns of crystallized decomposition matrices correspond to the canonical basis elements of the Fock space (see Specht). Consequently, the entries in these matrices are polynomials (inv), and by specializing (i.e. settingvequal to 1; see Specialized), the decomposition matrices of H are obtained (see Specht).Crystallized decomposition matrices are defined only for Hecke algebras over a base field of characteristic zero. Unlike ``normal'' decomposition matrices, crystallized decomposition matrices cannot be induced.
gap> CrystalDecompositionMatrix(Specht(3), 6); 6 | 1 5,1 | v 1 4,2 | . . 1 4,1^2 | . v . 1 3^2 | . v . . 1 3,2,1 | v v^2 . v v 1 3,1^3 | . . . v^2 . v 2^3 | v^2 . . . . v 2^2,1^2| . . . . . . 1 2,1^4 | . . . . v v^2 . 1^6 | . . . . v^2 . . gap> Specialized(last); # set 'v' equal to $1$. 6 | 1 5,1 | 1 1 4,2 | . . 1 4,1^2 | . 1 . 1 3^2 | . 1 . . 1 3,2,1 | 1 1 . 1 1 1 3,1^3 | . . . 1 . 1 2^3 | 1 . . . . 1 2^2,1^2| . . . . . . 1 2,1^4 | . . . . 1 1 . 1^6 | . . . . 1 . .See also
SpechtSpecht,SchurSchur,DecompositionMatrixDecompositionMatrix, andSpecializedSpecialized. This function requires the package ``specht'' (see RequirePackage).
DecompositionNumber(H, mu, nu)
DecompositionNumber(d, mu, nu)This function attempts to calculate the decomposition multiplicity of
D(nu) inS(mu) (equivalently, the multiplicity ofS(mu) inP(nu)). IfP(nu) is known, we just look up the answer; if notDecompositionNumbertries to calculate the answer using ``row and column removal'' (see [J,Theorem~6.18]).
gap> H:=Specht(6);; gap> DecompositionNumber(H,[6,4,2],[6,6]); 0This function requires the package ``specht'' (see RequirePackage).
InducedModule(x)
InducedModule(x, r_1 [,r_2, ...])There is an natural embedding of 'H'(Sym_n) in 'H'(Sym_{n+1}) which in the usual way lets us define an induced 'H'(Sym_{n+1})--module for every 'H'(Sym_n)--module. The function
InducedModulereturns the induced modules of the Specht modules, principal indecomposable modules, and simple modules (more accurately, their image in the Grothendieck ring).There is also a function
SInducedModule(see SInducedModule) which provides a much faster way of r--inducing s times (and inducing s times).Let mu be a partition. Then the induced module
InducedModule(S(mu))is easy to describe: it has the same composition factors as sum 'S'(<nu>) where nu runs over all partitions whose diagrams can be obtained by adding a single node to the diagram of mu.
gap> H:=Specht(2,2); Specht(e=2, p=2, S(), P(), D()) gap> InducedModule(H.S(7,4,3,1)); S(8,4,3,1)+S(7,5,3,1)+S(7,4,4,1)+S(7,4,3,2)+S(7,4,3,1,1) gap> InducedModule(H.P(5,3,1)); P(6,3,1)+2*P(5,4,1)+P(5,3,2) gap> InducedModule(H.D(11,2,1)); # D(<x>), unable to rewrite <x> as a sum of simples S(12,2,1)+S(11,3,1)+S(11,2,2)+S(11,2,1,1)When inducing indecomposable modules and simple modules,
InducedModulefirst rewrites these modules as a linear combination of Specht modules (using known decomposition matrices), and then induces this linear combination of Specht modules. If possible Specht then rewrites the induced module back in the original basis. Note that in the last example above, the decomposition matrix for Sym_{15} is not known by Specht; this is whyInducedModulewas unable to rewrite this module in theD--basis.r--Induction
InducedModule(x, r_1 [, r_2, ...])Two Specht modules
S(mu) andS(nu) belong to the same block if and only if the corresponding partitions mu and nu have the same e--core [JM2] (see ECore). Because the e--core of a partition is determined by its (multiset of) e--residues, ifS(mu) andS(nu) appear inInducedModule(S(tau)), for some partition tau, thenS(mu) andS(nu) belong to the same block if and only if mu and nu can be obtained by adding a node of the same e--residue to the diagram of tau. The second form ofInducedModuleallows one to induce ``within blocks'' by only adding nodes of some fixed e--residue r; this is known as r-induction. Note that 0 le r<e.
gap> H:=Specht(4); InducedModule(H.S(5,2,1)); S(6,2,1)+S(5,3,1)+S(5,2,2)+S(5,2,1,1) gap> InducedModule(H.S(5,2,1),0); 0*S() gap> InducedModule(H.S(5,2,1),1); S(6,2,1)+S(5,3,1)+S(5,2,1,1) gap> InducedModule(H.S(5,2,1),2); 0*S() gap> InducedModule(H.S(5,2,1),3); S(5,2,2)The function
EResidueDiagram(EResidueDiagram), prints the diagram of mu, labeling each node with its e--residue. A quick check of this diagram confirms the answers above.
gap> EResidueDiagram(H,5,2,1); 0 1 2 3 0 3 0 2``Quantized'' induction
When
InducedModuleis applied to the canonical basis elementsH.Pq(mu) (or more generally elements of the Fock space; see Specht), a ``quantum analogue'' of induction is applied. More precisely, the functionInducedModule(*,i)corresponds to the action of the generator F_i of the quantum group U_q(widehat{sl_e}) on F[LLT].
gap> H:=Specht(3);; InducedModule(H.Pq(4,2),1,2); S(6,2)+v*S(4,4)+v^2*S(4,2,2) gap> H.P(last); P(6,2)See also
SInducedModuleSInducedModule,RestrictedModuleRestrictedModule, andSRestrictedModuleSRestrictedModule. This function requires the package ``specht'' (see RequirePackage).
SInducedModule(x, s)
SInducedModule(x, s, r)The function
SInducedModule, standing for ``string induction'', provides a more efficient way of r--inducing s times (and a way of inducing s times if the residue r is omitted); r--induction is explained in InducedModule.
gap> H:=Specht(4);; SInducedModule(H.P(5,2,1),3); P(8,2,1)+3*P(7,3,1)+2*P(7,2,2)+6*P(6,3,2)+6*P(6,3,1,1)+3*P(6,2,1,1,1) +2*P(5,3,3)+P(5,2,2,1,1) gap> SInducedModule(H.P(5,2,1),3,1); P(6,3,1,1) gap> InducedModule(H.P(5,2,1),1,1,1); 6*P(6,3,1,1)Note that the multiplicity of each summand of
InducedModule(x,r,...,r)is divisible by <s>! and thatSInducedModuledivides by this constant.As with
InducedModulethis function can also be applied to elements of the Fock space (see Specht), in which case the quantum analogue of induction is used.See also
InducedModuleInducedModule. This function requires the package ``specht'' (see RequirePackage).
RestrictedModule(x)
RestrictedModule(x, r_1 [, r_2, ...])
Given a module x for 'H'(Sym_n)
RestrictedModulereturns the corresponding module for 'H'(Sym_{n-1}). The restriction of the Specht moduleS(mu) is the linear combination of Specht modules sum 'S'(<nu>) where nu runs over the partitions whose diagrams are obtained by deleting a node from the diagram of mu. If only nodes of residue r are deleted then this corresponds to first restrictingS(mu) and then taking one of the block components of the restriction; this process is known as r-restriction (cf. r--induction in InducedModule).There is also a function
SRestrictedModule(see SRestrictedModule) which provides a faster way of r--restricting s times (and restricting s times).When more than one residue if given to
RestrictedModuleit returnsRestrictedModule(x,r_1,r_2,...,r_k)=RestrictedModule(RestrictedModule(x,r_1),r_2,...,r_k) (cf.InducedModuleInducedModule).
gap> H:=Specht(6);; RestrictedModule(H.P(5,3,2,1),4); 2*P(4,3,2,1) gap> RestrictedModule(H.D(5,3,2),1); D(5,2,2)``Quantized'' restriction
As with
InducedModule, whenRestrictedModuleis applied to the canonical basis elementsH.Pq(mu) a quantum analogue of restriction is applied; this time,RestrictedModule(*,i)corresponds to the action of the generator E_i of U_q(widehat{sl_e}) on F[LLT].See also
InducedModuleInducedModule,SInducedModuleSInducedModule, andSRestrictedModuleSRestrictedModule. This function requires the package ``specht'' (see RequirePackage).
SRestrictedModule(x, s)
SRestrictedModule(x, s, r)As with
SInducedModulethis function provides a more efficient way of r--restricting s times, or restricting s times if the residue r is omitted (cf.SInducedModuleSInducedModule).
gap> H:=Specht(6);; SRestrictedModule(H.S(4,3,2),3); 3*S(4,2)+2*S(4,1,1)+3*S(3,3)+6*S(3,2,1)+2*S(2,2,2) gap> SRestrictedModule(H.P(5,4,1),2,4); P(4,4)See also
InducedModuleInducedModule,SInducedModuleSInducedModule, andRestrictedModuleRestrictedModule. This function requires the package ``specht'' (see RequirePackage).
71.10 InducedDecompositionMatrix
InducedDecompositionMatrix(d)If d is the decomposition matrix of 'H'(Sym_n), then
InducedDecompositionMatrix(d)attempts to calculate the decomposition matrix of 'H'(Sym_{n+1}). It does this by extracting each projective indecomposable from d and inducing these modules to obtain projective modules for 'H'(Sym_{n+1}).InducedDecompositionMatrixthen tries to decompose these projectives using the functionIsNewIndecomposable(see IsNewIndecomposable). In general there will be columns of the decomposition matrix whichInducedDecompositionMatrixis unable to decompose and these will have to be calculated ``by hand''.InducedDecompositionMatrixprints a list of those columns of the decomposition matrix which it is unable to calculate (this list is also printed by the functionMissingIndecomposables(d)).
gap> gap> d:=DecompositionMatrix(Specht(3,3),14);; gap> InducedDecompositionMatrix(d);; # Inducing.... The following projectives are missing from <d>: [ 15 ] [ 8, 7 ]Note that the missing indecomposables come in ``pairs'' which map to each other under the Mullineux map (see
MullineuxMapMullineuxMap).Almost all of the decomposition matrices included in Specht were calculated directly by
InducedDecompositionMatrix. When n is ``small''InducedDecompositionMatrixis usually able to return the full decomposition matrix for 'H'(Sym_{n+1}).Finally, although the
InducedDecompositionMatrixcan also be applied to the decomposition matrices of the q--Schur algebras (seeSchurSchur),InducedDecompositionMatrixis much less successful in inducing these decomposition matrices because it contains no special routines for dealing with the indecomposable modules of the q--Schur algebra which are indexed by e--singular partitions. Note also that we use a non--standard labeling of the decomposition matrices of q--Schur algebras; see Schur.
IsNewIndecomposable(d, x [,mu])
IsNewIndecomposableis the function which does all of the hard work when the functionInducedDecompositionMatrixis applied to decomposition matrices (see InducedDecompositionMatrix). Given a projective module x,IsNewIndecomposablereturnstrueif it is able to show that x is indecomposable (and this indecomposable is not already listed in d), andfalseotherwise.IsNewIndecomposablewill also print a brief description of its findings, giving an upper and lower bound on the first decomposition number mu for which it is unable to determine the multiplicity ofS(mu) in x.
IsNewIndecomposableworks by running through all of the partitions nu such thatP(nu) could be a summand of x and it uses various results, such as the q-Schaper theorem of [JM2] (seeSchaperSchaper), the Mullineux map (seeMullineuxMapMullineuxMap), and inducing simple modules, to determine ifP(nu) does indeed split off. In addition, if d is the decomposition matrix for 'H'(Sym_n) thenIsNewIndecomposablewill probably use some of the decomposition matrices of 'H'(Sym_m) for m le n, if they are known. Consequently it is a good idea to save decomposition matrices as they are calculated (see SaveDecompositionMatrix).For example, in calculating the 2--modular decomposition matrices of Sym_{r} the first projective which
InducedDecompositionMatrixis unable to calculate isP(10).
gap> H:=Specht(2,2);; gap> d:=InducedDecompositionMatrix(DecompositionMatrix(H,9));; # Inducing. # The following projectives are missing from <d>: # [ 10 ](In fact, given the above commands, Specht will return the full decomposition matrix for Sym_{10} because this matrix is in the library; these were the commands that I used to calculate the decomposition matrix in the library.)
By inducing
P(9) we can find a projective H--module which containsP(10). We can then useIsNewIndecomposableto try and decompose this induced module into a sum of PIMs.
gap> SpechtPrettyPrint(); x:=InducedModule(H.P(9),1); S(10)+S(9,1)+S(8,2)+2S(8,1^2)+S(7,3)+2S(7,1^3)+3S(6,3,1)+3S(6,2^2) +4S(6,2,1^2)+2S(6,1^4)+4S(5,3,2)+5S(5,3,1^2)+5S(5,2^2,1)+2S(5,1^5) +2S(4^2,2)+2S(4^2,1^2)+2S(4,3^2)+5S(4,3,1^3)+2S(4,2^3)+5S(4,2^2,1^2) +4S(4,2,1^4)+2S(4,1^6)+2S(3^3,1)+2S(3^2,2^2)+4S(3^2,2,1^2) +3S(3^2,1^4)+3S(3,2^2,1^3)+2S(3,1^7)+S(2^3,1^4)+S(2^2,1^6)+S(2,1^8) +S(1^10) gap> IsNewIndecomposable(d,x); # The multiplicity of S(6,3,1) in P(10) is at least 1 and at most 2. false gap> x; S(10)+S(9,1)+S(8,2)+2S(8,1^2)+S(7,3)+2S(7,1^3)+2S(6,3,1)+2S(6,2^2) +3S(6,2,1^2)+2S(6,1^4)+3S(5,3,2)+4S(5,3,1^2)+4S(5,2^2,1)+2S(5,1^5) +2S(4^2,2)+2S(4^2,1^2)+2S(4,3^2)+4S(4,3,1^3)+2S(4,2^3)+4S(4,2^2,1^2) +3S(4,2,1^4)+2S(4,1^6)+2S(3^3,1)+2S(3^2,2^2)+3S(3^2,2,1^2) +2S(3^2,1^4)+2S(3,2^2,1^3)+2S(3,1^7)+S(2^3,1^4)+S(2^2,1^6)+S(2,1^8) +S(1^10)Notice that some of the coefficients of the Specht modules in x have changed; this is because
IsNewIndecomposablewas able to determine that the multiplicity ofS(6,3,1) was at most 2 and so it subtracted one copy ofP(6,3,1) from x.In this case, the multiplicity of
S(6,3,1) inP(10) is easy to resolve because general theory says that this multiplicity must be odd. Therefore, x-'P'(6,3,1) is projective. After subtractingP(6,3,1) from x we again useIsNewIndecomposableto see if x is now indecomposable. We can tellIsNewIndecomposablethat all of the multiplicities up to and includingS(6,3,1) have already been checked by giving it the addition argument mu=[6,3,1].
gap> x:=x-H.P(d,6,3,1);; IsNewIndecomposable(d,x,6,3,1); trueConsequently, <x>='P'(10) and we add it to the decomposition matrix d (and save it).
gap> AddIndecomposable(d,x); SaveDecompositionMatrix(d);A full description of what
IsNewIndecomposabledoes can be found by reading the comments inspecht.g. Any suggestions or improvements on this function would be especially welcome.See also
DecompositionMatrixDecompositionMatrix andInducedDecompositionMatrixInducedDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
71.12 InvertDecompositionMatrix
InvertDecompositionMatrix(d)Returns the inverse of the (e--regular part of) d, where d is a decomposition matrix, or crystallized decomposition matrix, of a Hecke algebra or q--Schur algebra. If part of the decomposition matrix d is unknown then
InvertDecompositionMatrixwill invert as much of d as possible.
gap> H:=Specht(4);; d:=CrystalDecompositionMatrix(H,5);; gap> InvertDecompositionMatrix(d); 5 | 1 4,1 | . 1 3,2 | -v . 1 3,1^2| . . . 1 2^2,1| v^2 . -v . 1 2,1^3| . . . . . 1See also
DecompositionMatrixDecompositionMatrix, andCrystalDecompositionMatrixCrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
AdjustmentMatrix(dp, d)James [J] noticed, and Geck [G] proved, that the decomposition matrices dp for Hecke algebras defined over fields of positive characteristic admit a factorization dp = d * a where d is a decomposition matrix for a suitable Hecke algebra defined over a field of characteristic zero, and a is the so--called adjustment matrix. This function returns the adjustment matrix a.
gap> H:=Specht(2);; Hp:=Specht(2,2);; gap> d:=DecompositionMatrix(H,13);; dp:=DecompositionMatrix(Hp,13);; gap> a:=AdjustmentMatrix(dp,d); 13 | 1 12,1 | . 1 11,2 | 1 . 1 10,3 | . . . 1 10,2,1 | . . . . 1 9,4 | 1 . 1 . . 1 9,3,1 | 2 . . . . . 1 8,5 | . 1 . . . . . 1 8,4,1 | 1 . . . . . . . 1 8,3,2 | . 2 . . . . . 1 . 1 7,6 | 1 . . . . 1 . . . . 1 7,5,1 | . . . . . . 1 . . . . 1 7,4,2 | 1 . 1 . . 1 . . . . 1 . 1 7,3,2,1| . . . . . . . . . . . . . 1 6,5,2 | . 1 . . . . . 1 . 1 . . . . 1 6,4,3 | 2 . . . 1 . . . . . . . . . . 1 6,4,2,1| . 2 . 1 . . . . . . . . . . . . 1 5,4,3,1| 4 . 2 . . . . . . . . . . . . . . 1 gap> MatrixDecompositionMatrix(dp)= > MatrixDecompositionMatrix(d)*MatrixDecompositionMatrix(a); trueIn the last line we have checked our calculation.
See also
DecompositionMatrixDecompositionMatrix, andCrystalDecompositionMatrixCrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
SaveDecompositionMatrix(d)
SaveDecompositionMatrix(d, filename)The function
SaveDecompositionMatrixsaves the decomposition matrix d. After a decomposition matrix has been saved, the functionsH.S,H.P, andH.Dwill automatically access it as needed. So, for example, before saving d in order to retrieve the indecomposableP(mu) from d it is necessary to typeH.P(d, mu); once d has been saved, the commandH.P(mu)suffices.Since
InducedDecompositionMatrix(d)consults the decomposition matrices for smaller n, if they are available, it is advantageous to save decomposition matrices as they are calculated. For example, over a field of characteristic~5, the decomposition matrices for the symmetric groups Sym_n with n le 20 can be calculated as follows:
gap> H:=Specht(5,5);; gap> d:=DecompositionMatrix(H,9);; gap> for r in [10..20] do > d:=InducedDecompositionMatrix(d); > SaveDecompositionMatrix(d); > od;If your Hecke algebra record
His defined using a non--standard valuation map (see Specht) then it is also necessary to set the string ``H.HeckeRing'', or to supply the function with a filename before it will save your matrix.SaveDecompositionMatrixwill also save adjustment matrices and the various other matrices that appear in Specht (they can be read back in usingDecompositionMatrix). Each matrix has a default filename which you can over ride by supplying a filename. Using non--standard file names will stop Specht from automatically accessing these matrices in future.See also DecompositionMatrix
DecompositionMatrixDecompositionMatrix andCrystalDecompositionMatrixCrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
71.15 CalculateDecompositionMatrix
CalculateDecompositionMatrix(H,n)
CalculateDecompositionMatrix(H,n)is similar to the functionDecompositionMatrixDecompositionMatrix in that both functions try to return the decomposition matrixdof 'H'(Sym_n); the difference is that this function tries to calculate this matrix whereas the later reads the matrix from the library files (in characteristic zero both functions apply the algorithm of [LLT] to compute~d). In effect this function is only needed when working with Hecke algebras defined over fields of positive characteristic (or when you wish to avoid the libraries).For example, if you want to do calculations with the decomposition matrix of the symmetric group Sym_{15} over a field of characteristic two,
DecompositionMatrixreturns false whereasCalculateDecompositionMatrix; returns a part of the decomposition matrix.
gap> H:=Specht(2,2); Specht(e=2, p=2, S(), P(), D()) gap> d:=DecompositionMatrix(H,15); # This decomposition matrix is not known; use CalculateDecompositionMatrix() # or InducedDecompositionMatrix() to calculate with this matrix. false gap> d:=CalculateDecompositionMatrix(H,15);; # Projective indecomposable P(6,4,3,2) not known. # Projective indecomposable P(6,5,3,1) not known. ... gap> MissingIndecomposables(d); The following projectives are missing from <d>: [ 15 ] [ 14, 1 ] [ 13, 2 ] [ 12, 3 ] [ 12, 2, 1 ] [ 11, 4 ] [ 11, 3, 1 ] [ 10, 5 ] [ 10, 4, 1 ] [ 10, 3, 2 ] [ 9, 6 ] [ 9, 5, 1 ] [ 9, 4, 2 ] [ 9, 3, 2, 1 ] [ 8, 7 ] [ 8, 6, 1 ] [ 8, 5, 2 ] [ 8, 4, 3] [ 8, 4, 2, 1 ] [ 7, 6, 2 ] [ 7, 5, 3 ] [ 7, 5, 2, 1 ] [ 7, 4, 3, 1 ] [ 6, 5, 4 ] [ 6, 5, 3, 1 ] [ 6, 4, 3, 2 ]Actually, you are much better starting with the decomposition matrix of Sym_{14} and then applying
InducedDecompositionMatrixto this matrix.See also DecompositionMatrix
DecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
71.16 MatrixDecompositionMatrix
MatrixDecompositionMatrix(d)Returns the GAP matrix corresponding to the Specht decomposition matrix d. The rows and columns of d are ordered by
H.Ordering.
gap> MatrixDecompositionMatrix(DecompositionMatrix(Specht(3),5)); [ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 1, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ] ]See also
DecompositionMatrixDecompositionMatrix andDecompositionMatrixMatrixDecompositionMatrixMatrix. This function requires the package ``specht'' (see RequirePackage).
71.17 DecompositionMatrixMatrix
DecompositionMatrixMatrix(H, m, n)Given a Hecke algebra H, a GAP matrix m, and an integer n this function returns the Specht decomposition matrix corresponding to m. If
pis the number of partitions of n andrthe number of e--regular partitions of n, then m must be either <r>times<r>, <p>times<r>, or <p>times<p>. The rows and columns of m are assumed to be indexed by partitions ordered byH.Ordering(see Specht).
gap> H:=Specht(3);; gap> m:=[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 1, 0 ], > [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ];; gap> DecompositionMatrixMatrix(H,m,4); 4 | 1 3,1 | . 1 2^2 | 1 . 1 2,1^2| . . . 1 1^4 | . . 1 .See also
DecompositionMatrixDecompositionMatrix andMatrixDecompositionMatrixMatrixDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
AddIndecomposable(d, x)
AddIndecomposable(d, x)inserts the indecomposable module x into the decomposition matrix d. If d already contains the indecomposable x then a warning is printed. The functionAddIndecomposablealso calculatesMullineuxMap(x)(see MullineuxMap) and adds this indecomposable to d (or checks to see that it agrees with the corresponding entry of d if this indecomposable is already by d).See
IsNewIndecomposableIsNewIndecomposable for an example. See alsoDecompositionMatrixDecompositionMatrix andCrystalDecompositionMatrixCrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
RemoveIndecomposable(d, mu)The function
RemoveIndecomposableremoves the column from d which corresponds toP(mu). This is sometimes useful when trying to calculate a new decomposition matrix using Specht and want to test a possible candidate for a yet to be identified PIM.See also
DecompositionMatrixDecompositionMatrix andCrystalDecompositionMatrixCrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
MissingIndecomposables(d)The function
MissingIndecomposablesprints the list of partitions corresponding to the indecomposable modules which are not listed in d.See also
DecompositionMatrixDecompositionMatrix andCrystalDecompositionMatrixCrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
SimpleDimension(d)
SimpleDimension(H, n)
SimpleDimension(H|d, mu)In the first two forms,
SimpleDimensionprints the dimensions of all of the simple modules specified by d or for the Hecke algebra 'H'(Sym_n) respectively. If a partition mu is supplied, as in the last form, then the dimension of the simple moduleD(mu) is returned. At present the function is not implemented for the simple modules of the q--Schur algebras.
gap> H:=Specht(6);; gap> SimpleDimension(H,11,3); 272 gap> d:=DecompositionMatrix(H,5);; SimpleDimension(d,3,2); 5 gap> SimpleDimension(d); 5 : 1 4,1 : 4 3,2 : 5 3,1^2 : 6 2^2,1 : 5 2,1^3 : 4 1^5 : 1This function requires the package ``specht'' (see RequirePackage).
SpechtDimension(mu)Calculates the dimension of the Specht module
S(mu), which is equal to the number of standard mu-tableaux; the answer is given by the hook length formula (see [JK]).
gap> SpechtDimension(6,3,2,1); 5632See also
SimpleDimensionSimpleDimension. This function requires the package ``specht'' (see RequirePackage).
Schaper(H, mu)Given a partition mu, and a Hecke algebra H,
Schaperreturns a linear combination of Specht modules which have the same composition factors as the sum of the modules in the ``Jantzen filtration'' ofS(mu); see [JM2]. In particular, if nu strictly dominates mu thenD(nu) is a composition factor ofS(mu) if and only if it is a composition factor ofSchaper(mu).
Schaperuses the valuation mapH.valuationattached to H (see Specht and [JM2]).One way in which the q--Schaper theorem can be applied is as follows. Suppose that we have a projective module x, written as a linear combination of Specht modules, and suppose that we are trying to decide whether the projective indecomposable
P(mu) is a direct summand of x. Then, providing that we know thatP(nu) is not a summand of x for all (e--regular) partitions nu which strictly dominate mu (see Dominates),P(mu) is a summand of x if and only ifInnerProduct(Schaper(H,mu),x)is non--zero (note, in particular, that we don't need to know the indecomposableP(mu) in order to perform this calculation).The q--Schaper theorem can also be used to check for irreduciblity; in fact, this is the basis for the criterion employed by
IsSimpleModule.
gap> H:=Specht(2);; gap> Schaper(H,9,5,3,2,1); S(17,2,1)-S(15,2,1,1,1)+S(13,2,2,2,1)-S(11,3,3,2,1)+S(10,4,3,2,1)-S(9,8,3) -S(9,8,1,1,1)+S(9,6,3,2)+S(9,6,3,1,1)+S(9,6,2,2,1) gap> Schaper(H,9,6,5,2); 0*S(0)The last calculation shows that
S(9,6,5,2) is irreducible when R is a field of characteristic 0 ande=2(cf.IsSimpleModule(H,9,6,5,2)).This function requires the package ``specht'' (see RequirePackage).
IsSimpleModule(H, mu)mu an e--regular partition.
Given an e--regular partition mu,
IsSimpleModule(H, mu)returnstrueifS(mu) is simple andfalseotherwise. This calculation uses the valuation functionH.valuation; see Specht. Note that the criterion used byIsSimpleModuleis completely combinatorial; it is derived from the q--Schaper theorem [JM2].
gap> H:=Specht(3);; gap> IsSimpleModule(H,45,31,24); falseSee also
SchaperSchaper. This function requires the package ``specht'' (see RequirePackage).
MullineuxMap(e|H, mu)
MullineuxMap(d, mu)
MullineuxMap(x)Given an integer e, or a Specht record H, and a partition mu,
MullineuxMap(e, mu) returns the image of mu under the Mullineux map; which we now explain.The sign representation
D(1^n) of the Hecke algebra is the (one dimensional) representation sending T_w to (-1)^{ell(w)}. The Hecke algebra H is not a Hopf algebra so there is no well defined action of H upon the tensor product of two H--modules; however, there is an outer automorphism # of H which corresponds to tensoring withD(1^n). This sends an irreducible module 'D'(<mu>) to an irreducible 'D'(<mu>)^#cong 'D'(<mu^#>) for some e--regular partition mu^#. In the symmetric group case, Mullineux gave a conjectural algorithm for calculating mu^#; consequently the map sending mu to mu^# is known as the Mullineux map.Deep results of Kleshchev [K] for the symmetric group give another (proven) algorithm for calculating the partition mu^# (Ford and Kleshchev have deduced Mullineux's conjecture from this). Using the canonical basis, it was shown by [LLT] that the natural generalization of Kleshchev's algorithm to H gives the Mullineux map for Hecke algebras over fields of characteristic zero. The general case follows from this, so the Mullineux map is now known for all Hecke algebras.
Kleshchev's map is easy to describe; he proved that if gns is any good node sequence for mu, then the sequence obtained from gns by replacing each residue r by -rbmod e is a good node sequence for mu^# (see
GoodNodeSequenceGoodNodeSequence).
gap> MullineuxMap(Specht(2),12,5,2); [ 12, 5, 2 ] gap> MullineuxMap(Specht(4),12,5,2); [ 4, 4, 4, 2, 2, 1, 1, 1 ] gap> MullineuxMap(Specht(6),12,5,2); [ 4, 3, 2, 2, 2, 2, 2, 1, 1 ] gap> MullineuxMap(Specht(8),12,5,2); [ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 ] gap> MullineuxMap(Specht(10),12,5,2); [ 3, 3, 3, 3, 2, 1, 1, 1, 1, 1 ]
MullineuxMap(d, mu)The Mullineux map can also be calculated using a decomposition matrix. To see this recall that ``tensoring'' a Specht module
S(mu) with the sign representation yields a module isomorphic to the dual ofS(lambda), where lambda is the partition conjugate to mu. It follows that d_{munu}=d_{lambdanu^#} for all e--regular partitions nu. Therefore, if mu is the last partition in the lexicographic order such that d_{munu}ne0 then we must have nu^#=lambda. The second form ofMullineuxMapuses d to calculate mu^# rather than the Kleshchev-[LLT] result.
MullineuxMap(x)In the third form, x is a module, and
MullineuxMapreturns <x>^#, the image of x under #. Note that the above remarks show thatP(mu) is mapped toP(mu^#) via the Mullineux map; this observation is useful when calculating decomposition matrices (and is used by the functionInducedDecompositionMatrix).See also
GoodNodesGoodNodes andGoodNodeSequenceGoodNodeSequence . This function requires the package ``specht'' (see RequirePackage).
MullineuxSymbol(e|H, mu)Returns the Mullineux symbol of the e--regular partition mu.
gap> MullineuxSymbol(5,[8,6,5,5]); [ [ 10, 6, 5, 3 ], [ 4, 4, 3, 2 ] ]See also
PartitionMullineuxSymbolPartitionMullineuxSymbol. This function requires the package ``specht'' (see RequirePackage).
71.27 PartitionMullineuxSymbol
PartitionMullineuxSymbol(e|H, ms)Given a Mullineux symbol ms, this function returns the corresponding e--regular partition.
gap> PartitionMullineuxSymbol(5, MullineuxSymbol(5,[8,6,5,5]) ); [ 8, 6, 5, 5 ]See also
MullineuxSymbolMullineuxSymbol. This function requires the package ``specht'' (see RequirePackage).
GoodNodes(e|H, mu)
GoodNodes(e|H, mu, r)Given a partition and an integer e, Kleshchev [K] defined the notion of good node for each residue r (0 le r<e). When e is prime and mu is e--regular, Kleshchev showed that the good nodes describe the restriction of the socle of
D(mu) in the symmetric group case. Brundan~[B] has recently generalized this result to the Hecke algebra.By definition, there is at most one good node for each residue r, and this node is a removable node (in the diagram of mu). The function
GoodNodesreturns a list of the rows of mu which end in a good node; the good node of residue r (if it exists) is the (r+1)--st element in this list. In the second form, the number of the row which ends with the good node of residue r is returned; orfalseif there is no good node of residue r.
gap> GoodNodes(5,[5,4,3,2]); [ false, false, 2, false, 1 ] gap> GoodNodes(5,[5,4,3,2],0); false gap> GoodNodes(5,[5,4,3,2],4); 1The good nodes also determine the Kleshchev--Mullineux map (see
GoodNodeSequenceGoodNodeSequence andMullineuxMapMullineuxMap). This function requires the package ``specht'' (see RequirePackage).
NormalNodes(e|H, mu)
NormalNodes(e|H, mu, r)Returns the numbers of the rows of mu which end in one of Kleshchev's [K] normal nodes. In the second form, only those rows corresponding to normal nodes of the specified residue are returned.
gap> NormalNodes(5,[6,5,4,4,3,2,1,1,1]); [ [ 1, 4 ], [ ], [ ], [ 2, 5 ], [ ] ] gap> NormalNodes(5,[6,5,4,4,3,2,1,1,1],0); [ 1, 4 ]See also
GoodNodesGoodNodes. This function requires the package ``specht'' (see RequirePackage).
GoodNodeSequence(e|H, mu)
GoodNodeSequences(e|H, mu)mu an e--regular partition.
Given an e--regular partition mu of n, a good node sequence for mu is a sequence gns of n residues such that mu has a good node of residue r, where r is the last residue in gns, and the first n-1 residues in gns are a good node sequence for the partition obtained from mu by deleting its (unique) good node with residue r (see
GoodNodesGoodNodes). In general, mu will have more than one good node sequence; however, any good node sequence uniquely determines mu (seePartitionGoodNodeSequencePartitionGoodNodeSequence).
gap> H:=Specht(4);; GoodNodeSequence(H,4,3,1); [ 0, 3, 1, 0, 2, 2, 1, 3 ] gap> GoodNodeSequence(H,4,3,2); [ 0, 3, 1, 0, 2, 2, 1, 3, 3 ] gap> GoodNodeSequence(H,4,4,2); [ 0, 3, 1, 0, 2, 2, 1, 3, 3, 2 ] gap> GoodNodeSequence(H,5,4,2); [ 0, 3, 1, 0, 2, 2, 1, 3, 3, 2, 0 ]The function
GoodNodeSequencesreturns the list of all good node sequences for mu.
gap> GoodNodeSequences(H,5,2,1); [ [ 0, 1, 2, 3, 3, 2, 0, 0 ], [ 0, 3, 1, 2, 2, 3, 0, 0 ], [ 0, 1, 3, 2, 2, 3, 0, 0 ], [ 0, 1, 2, 3, 3, 0, 2, 0 ], [ 0, 1, 2, 3, 0, 3, 2, 0 ], [ 0, 1, 2, 3, 3, 0, 0, 2 ], [ 0, 1, 2, 3, 0, 3, 0, 2 ] ]The good node sequences determine the Mullineux map (see
GoodNodesGoodNodes andMullineuxMapMullineuxMap). This function requires the package ``specht'' (see RequirePackage).
71.31 PartitionGoodNodeSequence
PartitionGoodNodeSequence(e|H, gns)Given a good node sequence gns (see
GoodNodeSequenceGoodNodeSequence), this function returns the unique e--regular partition corresponding to gns (orfalseif in fact gns is not a good node sequence).
gap> H:=Specht(4);; gap> PartitionGoodNodeSequence(H,0, 3, 1, 0, 2, 2, 1, 3, 3, 2); [ 4, 4, 2 ]See also
GoodNodesGoodNodes,GoodNodeSequenceGoodNodeSequence andMullineuxMapMullineuxMap. This function requires the package ``specht'' (see RequirePackage).
GoodNodeLatticePath(e|H, mu)
GoodNodeLatticePaths(e|H, mu)
LatticePathGoodNodeSequence(e|H, gns)The function
GoodNodeLatticePathreturns a sequence of partitions which give a path in the e--good partition lattice from the empty partition to mu. The second function returns the list of all paths in the e--good partition lattice which end in mu, and the third function returns the path corresponding to a given good node sequence gns.
gap> GoodNodeLatticePath(3,3,2,1); [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 1, 1 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ] gap> GoodNodeLatticePaths(3,3,2,1); [ [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 1, 1 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ], [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 2 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ] ] gap> GoodNodeSequence(4,6,3,2); [ 0, 3, 1, 0, 2, 2, 3, 3, 0, 1, 1 ] gap> LatticePathGoodNodeSequence(4,last); [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 2 ], [ 3, 2 ], [ 3, 2, 1 ], [ 4, 2, 1 ], [ 4, 2, 2 ], [ 5, 2, 2 ], [ 6, 2, 2 ], [ 6, 3, 2 ] ]See also
GoodNodesGoodNodes. This function requires the package ``specht'' (see RequirePackage).
71.33 LittlewoodRichardsonRule
LittlewoodRichardsonRule(mu, nu)
LittlewoodRichardsonCoefficient(mu, nu, tau)Given partitions mu of n and nu of m the module 'S'(<mu>) otimes 'S'(<nu>) is naturally an 'H'(Sym_ntimesSym_m)-module and, by inducing, we obtain an 'H'(Sym_{n+m})-module. This module has the same composition factors as sum_nu a_munu^lambda
S(lambda), where the sum runs over all partitions lambda of n+m and the integers a_{munu}^lambda are the Littlewood--Richardson coefficients. The integers a_{munu}^lambda can be calculated using a straightforward combinatorial algorithm known as the Littlewood--Richardson rule (see [JK]).The function
LittlewoodRichardsonRulereturns an (unordered) list of partitions of n+m in which each partition lambda occurs a_{munu}^lambda times. The Littlewood-Richardson coefficients are independent of e; they can be read more easily from the computationS(mu)*S(nu).
gap> H:=Specht(0);; # the generic Hecke algebra with 'R'=*C*['q'] gap> LittlewoodRichardsonRule([3,2,1],[4,2]); [ [ 4, 3, 2, 2, 1 ],[ 4, 3, 3, 1, 1 ],[ 4, 3, 3, 2 ],[ 4, 4, 2, 1, 1 ], [ 4, 4, 2, 2 ],[ 4, 4, 3, 1 ],[ 5, 2, 2, 2, 1 ],[ 5, 3, 2, 1, 1 ], [ 5, 3, 2, 2 ],[ 5, 4, 2, 1 ],[ 5, 3, 2, 1, 1 ],[ 5, 3, 3, 1 ], [ 5, 4, 1, 1, 1 ],[ 5, 4, 2, 1 ],[ 5, 5, 1, 1 ],[ 5, 3, 2, 2 ], [ 5, 3, 3, 1 ],[ 5, 4, 2, 1 ],[ 5, 4, 3 ],[ 5, 5, 2 ],[ 6, 2, 2, 1, 1], [ 6, 3, 1, 1, 1 ],[ 6, 3, 2, 1 ],[ 6, 4, 1, 1 ],[ 6, 2, 2, 2 ], [ 6, 3, 2, 1 ],[ 6, 4, 2 ],[ 6, 3, 2, 1 ],[ 6, 3, 3 ],[ 6, 4, 1, 1 ], [ 6, 4, 2 ], [ 6, 5, 1 ], [ 7, 2, 2, 1 ], [ 7, 3, 1, 1 ], [ 7, 3, 2 ], [ 7, 4, 1 ] ] gap> H.S(3,2,1)*H.S(4,2); S(7,4,1)+S(7,3,2)+S(7,3,1,1)+S(7,2,2,1)+S(6,5,1)+2*S(6,4,2)+2*S(6,4,1,1) +S(6,3,3)+3*S(6,3,2,1)+S(6,3,1,1,1)+S(6,2,2,2)+S(6,2,2,1,1)+S(5,5,2) +S(5,5,1,1)+S(5,4,3)+3*S(5,4,2,1)+S(5,4,1,1,1)+2*S(5,3,3,1)+2*S(5,3,2,2) +2*S(5,3,2,1,1)+S(5,2,2,2,1)+S(4,4,3,1)+S(4,4,2,2)+S(4,4,2,1,1)+S(4,3,3,2) +S(4,3,3,1,1)+S(4,3,2,2,1) gap> LittlewoodRichardsonCoefficient([3,2,1],[4,2],[5,4,2,1]); 3The function
LittlewoodRichardsonCoefficientreturns a single Littlewood--Richardson coefficient (although you are really better off asking for all of them, since they will all be calculated anyway).See also
InducedModuleInducedModule andInverseLittlewoodRichardsonRuleInverseLittlewoodRichardsonRule. This function requires the package ``specht'' (see RequirePackage).
71.34 InverseLittlewoodRichardsonRule
InverseLittlewoodRichardsonRule(tau)Returns a list of all pairs of partitions [mu,nu] such that the Littlewood-Richardson coefficient a_{munu}^tau is non-zero (see LittlewoodRichardsonRule). The list returned is unordered and [mu,nu] will appear a_{munu}^tau times in it.
gap> InverseLittlewoodRichardsonRule([3,2,1]); [ [ [ ],[ 3, 2, 1 ] ],[ [ 1 ],[ 3, 2 ] ],[ [ 1 ],[ 2, 2, 1 ] ], [ [ 1 ],[ 3, 1, 1 ] ],[ [ 1, 1 ],[ 2, 2 ] ],[ [ 1, 1 ],[ 3, 1 ] ], [ [ 1, 1 ],[ 2, 1, 1 ] ],[ [ 1, 1, 1 ],[ 2, 1 ] ],[ [ 2 ],[ 2, 2 ] ], [ [ 2 ],[ 3, 1 ] ],[ [ 2 ],[ 2, 1, 1 ] ],[ [ 2, 1 ],[ 3 ] ], [ [ 2, 1 ],[ 2, 1 ] ],[ [ 2, 1 ],[ 2, 1 ] ],[ [ 2, 1 ],[ 1, 1, 1 ] ], [ [ 2, 1, 1 ],[ 2 ] ],[ [ 2, 1, 1 ],[ 1, 1 ] ],[ [ 2, 2 ],[ 2 ] ], [ [ 2, 2 ],[ 1, 1 ] ],[ [ 2, 2, 1 ],[ 1 ] ],[ [ 3 ],[ 2, 1 ] ], [ [ 3, 1 ],[ 2 ] ],[ [ 3, 1 ],[ 1, 1 ] ],[ [ 3, 1, 1 ],[ 1 ] ], [ [ 3, 2 ],[ 1 ] ],[ [ 3, 2, 1 ],[ ] ] ]See also
LittlewoodRichardsonRuleLittlewoodRichardsonRule.This function requires the package ``specht'' (see RequirePackage).
EResidueDiagram(H|e, mu)
EResidueDiagram(x)The e--residue of the (i,j)--th node in the diagram of a partition mu is (j-i)bmod <e>.
EResidueDiagram(e, mu)prints the diagram of the partition mu replacing each node with its e-residue.If x is a module then
EResidueDiagram(x)prints the e--residue diagrams of all of the e--regular partitions appearing in x (such diagrams are useful when trying to decide how to restrict and induce modules and also in applying results such as the ``Scattering theorem'' of [JM1]). It is not necessary to supply the integer e in this case because x ``knows'' the value of e.
gap> H:=Specht(2);; EResidueDiagram(H.S(H.P(7,5))); [ 7, 5 ] 0 1 0 1 0 1 0 1 0 1 0 1 [ 6, 5, 1 ] 0 1 0 1 0 1 1 0 1 0 1 0 [ 5, 4, 2, 1 ] 0 1 0 1 0 1 0 1 0 0 1 1 # There are 3 2-regular partitions.This function requires the package ``specht'' (see RequirePackage).
HookLengthDiagram(mu)Prints the diagram of mu, replacing each node with its hook length (see [JK]).
gap> HookLengthDiagram(11,6,3,2); 14 13 11 9 8 7 5 4 3 2 1 8 7 5 3 2 1 4 3 1 2 1This function requires the package ``specht'' (see RequirePackage).
RemoveRimHook(mu, row, col)Returns the partition obtained from mu by removing the (row, col)--th rim hook from (the diagram of) mu.
gap> RemoveRimHook([6,5,4],1,2); [ 4, 3, 1 ] gap> RemoveRimHook([6,5,4],2,3); [ 6, 3, 2 ] gap> HookLengthDiagram(6,5,4); 8 7 6 5 3 1 6 5 4 3 1 4 3 2 1See also
AddRimHookAddRimHook. This function requires the package ``specht'' (see RequirePackage).
AddRimHook(mu, r, h);Returns a list [nu, l] where nu is the partition obtained from mu by adding a rim hook of length h with its ``foot'' in the r--th row of (the diagram of) mu and l is the leg length of the wrapped on rim hook (see, for example,[JK]). If the resulting diagram nu is not the diagram of a partition then
falseis returned.
gap> AddRimHook([6,4,3],1,3); [ [ 9, 4, 3 ], 0 ] gap> AddRimHook([6,4,3],2,3); false gap> AddRimHook([6,4,3],3,3); [ [ 6, 5, 5 ], 1 ] gap> AddRimHook([6,4,3],4,3); [ [ 6, 4, 3, 3 ], 0 ] gap> AddRimHook([6,4,3],5,3); falseSee also
RemoveRimHookRemoveRimHook. This function requires the package ``specht'' (see RequirePackage).
ECore(H|e, mu)The e-core of a partition mu is what remains after as many rim e-hooks as possible have been removed from the diagram of mu (that this is well defined is not obvious; see [JK]). Thus,
ECore(mu)returns the e--core of the partition mu,
gap> H:=Specht(6);; ECore(H,16,8,6,5,3,1); [ 4, 3, 1, 1 ]The e--core is calculated here using James' notation of an abacus; there is also an
EAbacusfunction; but it is more ``pretty'' than useful. indexEAbacusSee also
IsECoreIsECore,EQuotientEQuotient, andEWeightEWeight. This function requires the package ``specht'' (see RequirePackage).
IsECore(H|e, mu)Returns
trueif mu is an e--core andfalseotherwise; seeECoreECore.See also
ECoreECore. This function requires the package ``specht'' (see RequirePackage).
EQuotient(H|e, mu)Returns the e-quotient of mu; this is a sequence of e partitions whose definition can be found in [JK].
gap> H:=Specht(8);; EQuotient(H,22,18,16,12,12,1,1); [ [ 1, 1 ], [ ], [ ], [ ], [ ], [ 2, 2 ], [ ], [ 1 ] ]See also
ECoreECore andCombineEQuotientECoreCombineEQuotientECore. This function requires the package ``specht'' (see RequirePackage).
CombineEQuotientECore(H|e, Q, C)A partition is uniquely determined by its e-quotient and its e-core (see EQuotient and ECore).
CombineEQuotientECore(e, Q, C)returns the partition which has e--quotient Q and e--core C. The integer e can be replaced with a record H which was created using the functionSpecht.
gap> H:=Specht(11);; mu:=[100,98,57,43,12,1];; gap> Q:=EQuotient(H,mu); [ [ 9 ], [ ], [ ], [ ], [ ], [ ], [ 3 ], [ 1 ], [ 9 ], [ ], [ 5 ] ] gap> C:=ECore(H,mu); [ 7, 2, 2, 1, 1, 1 ] gap> CombineEQuotientECore(H,Q,C); [ 100, 98, 57, 43, 12, 1 ]See also
ECoreECore andEQuotientEQuotient. This function requires the package ``specht'' (see RequirePackage).
EWeight(H|e, mu)The e--weight of a partition is the number of e--hooks which must be removed from the partition to reach the e--core (see
ECoreECore).
gap> EWeight(6,[16,8,6,5,3,1]); 5This function requires the package ``specht'' (see RequirePackage).
ERegularPartitions(H|e, n)A partition mu=(mu_1,mu_2,ldots) is e--regular if there is no integer i such that mu_i=mu_{i+1}=cdots=mu_{i+<e>-1}>0. The function
ERegularPartitions(e, n)returns the list of e--regular partitions of n, ordered reverse lexicographically (see Lexicographic).
gap> H:=Specht(3); Specht(e=3, S(), P(), D(), Pq()); gap> ERegularPartitions(H,6); [ [ 2, 2, 1, 1 ], [ 3, 2, 1 ], [ 3, 3 ], [ 4, 1, 1 ], [ 4, 2 ], [ 5, 1 ], [ 6 ] ]This function requires the package ``specht'' (see RequirePackage).
IsERegular(H|e, mu)Returns
trueif mu is e--regular andfalseotherwise.This functions requires the package ``specht'' (see RequirePackage).
ConjugatePartition(mu)Given a partition mu,
ConjugatePartition(mu)returns the partition whose diagram is obtained by interchanging the rows and columns in the diagram of mu.
gap> ConjugatePartition(6,4,3,2); [ 4, 4, 3, 2, 1, 1 ]This function requires the package ``specht'' (see RequirePackage).
BetaSet(mu)This function returns a set of beta numbers (i.e. first column hook lengths; see [JK]) corresponding to the partition mu.
gap> BetaSet([5,4,2,2]); [ 2, 3, 6, 8 ]See also
PartitionBetaSetPartitionBetaSet. This function requires the package ``specht'' (see RequirePackage).
PartitionBetaSet(bn)Given a set of beta numbers bn (see
BetaSetBetaSet), this function returns the corresponding partition. Note in particular that bn must be a set of integers.
gap> PartitionBetaSet([ 2, 3, 6, 8 ]); [ 5, 4, 2, 2 ]This function requires the package ``specht'' (see RequirePackage).
ETopLadder(H|e, mu)The ladders in the diagram of a partition are the lines connecting nodes of constant e--residue, having slope <e>-1 (see [JK]). A new partition can be obtained from mu by sliding all nodes up to the highest possible rungs on their ladders.
ETopLadder(e, mu)returns the partition obtained in this way; it is automatically e--regular (this partition is denoted mu^R in [JK]).
gap> H:=Specht(4);; gap> ETopLadder(H,1,1,1,1,1,1,1,1,1,1); [ 4, 3, 3 ] gap> ETopLadder(6,1,1,1,1,1,1,1,1,1,1); [ 2, 2, 2, 2, 2 ]This function requires the package ``specht'' (see RequirePackage).
Dominates(mu, nu)The dominance ordering is an important partial order in the representation theory of Hecke algebra because d_{munu}=0 unless nu dominates mu.
Dominates(mu, nu)returnstrueif either mu=nu or for all i ge 1, sum_{j=1}^imu_j ge sum_{j=1}^inu_j, andfalseotherwise.
gap> Dominates([5,4],[4,4,1]); trueThis function requires the package ``specht'' (see RequirePackage).
LengthLexicographic(mu, nu)
LengthLexicographicreturnstrueif the length of mu is less than the length of nu or if the length of mu equals the length of nu andLexicographic(mu, nu).
gap> p:=Partitions(6);;Sort(p,LengthLexicographic); p; [ [ 6 ],[ 5, 1 ],[ 4, 2 ],[ 3, 3 ],[ 4, 1, 1 ],[ 3, 2, 1 ],[ 2, 2, 2 ], [ 3, 1, 1, 1 ],[ 2, 2, 1, 1 ],[ 2, 1, 1, 1, 1 ],[ 1, 1, 1, 1, 1, 1 ] ]This function requires the package ``specht'' (see RequirePackage).
Lexicographic(mu, nu)
Lexicographic(mu, nu)returnstrueif mu is lexicographically greater than or equal to nu.
gap> p:=Partitions(6);;Sort(p,Lexicographic); p; [ [ 6 ],[ 5, 1 ],[ 4, 2 ],[ 4, 1, 1 ],[ 3, 3 ],[ 3, 2, 1 ], [ 3, 1, 1, 1 ],[ 2, 2, 2 ],[ 2, 2, 1, 1 ],[ 2, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1 ] ]This function requires the package ``specht'' (see RequirePackage).
ReverseDominance(mu, nu)This is another total order on partitions which extends the dominance ordering (see Dominates). Here mu is greater than nu if for all i>0 sum_jge imu_j sum_jge inu_j.
gap> p:=Partitions(6);;Sort(p,ReverseDominance); p; [ [ 6 ], [ 5, 1 ], [ 4, 2 ], [ 3, 3 ], [ 4, 1, 1 ], [ 3, 2, 1 ], [ 2, 2, 2 ], [ 3, 1, 1, 1 ], [ 2, 2, 1, 1 ], [ 2, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1 ] ]This is the ordering used by James in the appendix of his Springer lecture notes book.
This function requires the package ``specht'' (see RequirePackage).
Specialized(x [,q]);
Specialized(d [,q]);Given an element of the Fock space x (see Specht), or a crystallized decomposition matrix (see CrystalDecompositionMatrix),
Specializedreturns the corresponding element of the Grothendieck ring or the corresponding decomposition matrix of the Hecke algebra respectively. By default the indeterminatevis specialized to 1; howevervcan be specialized to any (integer) q by supplying a second argument.
gap> H:=Specht(2);; x:=H.Pq(6,2); S(6,2)+v*S(6,1,1)+v*S(5,3)+v^2*S(5,1,1,1)+v*S(4,3,1)+v^2*S(4,2,2) +(v^3 + v)*S(4,2,1,1)+v^2*S(4,1,1,1,1)+v^2*S(3,3,1,1)+v^3*S(3,2,2,1) +v^3*S(3,1,1,1,1,1)+v^3*S(2,2,2,1,1)+v^4*S(2,2,1,1,1,1) gap> Specialized(x); S(6,2)+S(6,1,1)+S(5,3)+S(5,1,1,1)+S(4,3,1)+S(4,2,2) +2*S(4,2,1,1)+S(4,1,1,1,1)+S(3,3,1,1)+S(3,2,2,1)+S(3,1,1,1,1,1) +S(2,2,2,1,1)+S(2,2,1,1,1,1) gap> Specialized(x,2); S(6,2)+2*S(6,1,1)+2*S(5,3)+4*S(5,1,1,1)+2*S(4,3,1)+4*S(4,2,2)+10*S(4,2,1,1) +4*S(4,1,1,1,1)+4*S(3,3,1,1)+8*S(3,2,2,1)+8*S(3,1,1,1,1,1)+8*S(2,2,2,1,1) +16*S(2,2,1,1,1,1)An example of
Specializebeing applied to a crystallized decomposition matrix can be found in CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).
ERegulars(x)
ERegulars(d)
ListERegulars(x)
ERegulars(x)prints a list of the e--regular partitions, together with multiplicities, which occur in the module x.ListERegulars(x)returns an actual list of these partitions rather than printing them.
gap> H:=Specht(8);; gap> x:=H.S(InducedModule(H.P(8,5,3)) ); S(9,5,3)+S(8,6,3)+S(8,5,4)+S(8,5,3,1)+S(6,5,3,3)+S(5,5,4,3)+S(5,5,3,3,1) gap> ERegulars(x); [ 9, 5, 3 ] [ 8, 6, 3 ] [ 8, 5, 4 ] [ 8, 5, 3, 1 ] [ 6, 5, 3, 3 ] [ 5, 5, 4, 3 ] [ 5, 5, 3, 3, 1 ] gap> H.P(x); P(9,5,3)+P(8,6,3)+P(8,5,4)+P(8,5,3,1)This example shows why these functions are useful: given a projective module x, as above, and the list of e--regular partitions in x we know the possible indecomposable direct summands of x.
Note that it is not necessary to specify what e is when calling this function because x ``knows'' the value of e.
The function
ERegularscan also be applied to a decomposition matrix d; in this case it returns the unitriangular submatrix of d whose rows and columns are indexed by the e--regular partitions.These function requires the package ``specht'' (see RequirePackage).
SplitECores(x)
SplitECores(x, mu)
SplitECores(x, y)The function
SplitECores(x)returns a list[b_1,...,b_k]where the Specht modules in each b_i all belong to the same block (i.e. they have the same e-core). Similarly,SplitECores(x, mu)returns the component of x which is in the same block as mu, andSplitECores(x, y)returns the component of x which is in the same block as y.
gap> H:=Specht(2);; gap> SplitECores(InducedModule(H.S(5,3,1))); [ S(6,3,1)+S(5,3,2)+S(5,3,1,1), S(5,4,1) ] gap> InducedModule(H.S(5,3,1),0); S(5,4,1) gap> InducedModule(H.S(5,3,1),1); S(6,3,1)+S(5,3,2)+S(5,3,1,1)See also
ECoreECore,InducedModuleInducedModule, andRestrictedModuleRestrictedModule.This function requires the package ``specht'' (see RequirePackage).
Coefficient(x, mu)
If x is a sum of Specht (resp. simple, or indecomposable) modules, then
Coefficient(x, mu)returns the coefficient ofS(mu) in x (resp.D(mu), orP(mu)).
gap> H:=Specht(3);; x:=H.S(H.P(7,3)); S(7,3)+S(7,2,1)+S(6,2,1^2)+S(5^2)+S(5,2^2,1)+S(4^2,1^2)+S(4,3^2)+S(4,3,2,1) gap> Coefficient(x,5,2,2,1); 1This function requires the package ``specht'' (see RequirePackage).
InnerProduct(x, y)Here x and y are some modules of the Hecke algebra (i.e. Specht modules, PIMS, or simple modules).
InnerProduct(x, y)computes the standard inner product of these elements. This is sometimes a convenient way to compute decomposition numbers (for example).
gap> InnerProduct(H.S(2,2,2,1), H.P(4,3)); 1 gap> DecompositionNumber(H,[2,2,2,1],[4,3]); 1This function requires the package ``specht'' (see RequirePackage).
SpechtPrettyPrint(true)
SpechtPrettyPrint(false)
SpechtPrettyPrint()This function changes the way in which Specht prints modules. The first two forms turn pretty printing on and off respectively (by default it is off), and the third form toggles the printing format.
gap> H:=Specht(2);; x:=H.S(H.P(6));; gap> SpechtPrettyPrint(true); x; S(6)+S(5,1)+S(4,1^2)+S(3,1^3)+S(2,1^4)+S(1^6) gap> SpechtPrettyPrint(false); x; S(6)+S(5,1)+S(4,1,1)+S(3,1,1,1)+S(2,1,1,1,1)+S(1,1,1,1,1,1) gap> SpechtPrettyPrint(); x; S(6)+S(5,1)+S(4,1^2)+S(3,1^3)+S(2,1^4)+S(1^6)This function requires the package ``specht'' (see RequirePackage).
SemiStandardTableaux(mu, nu)mu a partition, nu a composition.
Returns a list of the semistandard mu--tableaux of type nu[JK]. Tableaux are represented as lists of lists, with the first element of the list being the first row of the tableaux and so on.
gap> SemiStandardTableaux([4,3],[1,1,1,2,2]); [ [ [ 1, 2, 3, 4 ], [ 4, 5, 5 ] ], [ [ 1, 2, 3, 5 ], [ 4, 4, 5 ] ], [ [ 1, 2, 4, 4 ], [ 3, 5, 5 ] ], [ [ 1, 2, 4, 5 ], [ 3, 4, 5 ] ], [ [ 1, 3, 4, 4 ], [ 2, 5, 5 ] ], [ [ 1, 3, 4, 5 ], [ 2, 4, 5 ] ] ]See also
StandardTableauxStandardTableaux. This function requires the package ``specht'' (see RequirePackage).
StandardTableaux(mu)mu a partition.
Returns a list of the standard mu--tableaux.
gap> StandardTableaux(4,2); [ [ [ 1, 2, 3, 4 ], [ 5, 6 ] ], [ [ 1, 2, 3, 5 ], [ 4, 6 ] ], [ [ 1, 2, 3, 6 ], [ 4, 5 ] ], [ [ 1, 2, 4, 5 ], [ 3, 6 ] ], [ [ 1, 2, 4, 6 ], [ 3, 5 ] ], [ [ 1, 2, 5, 6 ], [ 3, 4 ] ], [ [ 1, 3, 4, 5 ], [ 2, 6 ] ], [ [ 1, 3, 4, 6 ], [ 2, 5 ] ], [ [ 1, 3, 5, 6 ], [ 2, 4 ] ] ]See also
SemiStandardTableauxSemiStandardTableaux. This function requires the package ``specht'' (see RequirePackage).
ConjugateTableau(tab)Returns the tableau obtained from tab by interchangings its rows and columns.
gap> ConjugateTableau([ [ 1, 3, 5, 6 ], [ 2, 4 ] ]); [ [ 1, 2 ], [ 3, 4 ], [ 5 ], [ 6 ] ]This function requires the package ``specht'' (see RequirePackage).
ShapeTableau(tab)Given a tableau tab this function returns the partition (or composition).
gap> ShapeTableau( [ [ 1, 1, 2, 3 ], [ 4, 5 ] ] ); [ 4, 2 ]This function requires the package ``specht'' (see RequirePackage).
TypeTableau(tab)Returns the type of the (semistandard) tableau tab; that is, the composition sigma=(sigma_1,sigma_2,ldots) where sigma_i is the number of entries in tab which are equal to i.
gap> List(SemiStandardTableaux([5,4,2],[4,3,0,1,3]),TypeTableau); [ [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ] ]This function requires the package ``specht'' (see RequirePackage).
Specht 2.4
September 1997