University of Sydney Algebra Seminar

Anthony Henderson (University of Sydney)

Friday 13 May, 12:05-12:55pm, Carslaw 175

Mirabolic and exotic Robinson-Schensted correspondences

The Robinson-Schensted correspondence is an important bijection between the symmetric group $S_n$ and the set of pairs of standard Young tableaux of the same shape with $n$ boxes. By fixing one of the tableaux and letting the other vary, one obtains the left and right cells in the symmetric group. The correspondence can be defined by a simple combinatorial algorithm, but it also has a nice geometric interpretation due to Steinberg. $S_n$ parametrizes the orbits of $GL(V)$ in $Fl(V)\times Fl(V)$, where $Fl(V)$ is the variety of complete flags in the vector space $V$ of dimension $n$. The conormal bundle to an orbit $O_w$ consists of triples $(F_1,F_2,x)$ where $(F_1,F_2)$ is in $O_w$ and $x$ is a nilpotent endomorphism of $V$ which preserves both flags. The tableaux corresponding to $w$ record the action of $x$ on $F_1$ and $F_2$ for a generic triple in this conormal bundle.

Roman Travkin gave a mirabolic generalization of the Robinson-Schensted correspondence, by considering the orbits of $GL(V)$ in $V\times Fl(V)\times Fl(V)$. Here $S_n$ is replaced by the set of marked permutations $(w,I)$ where $w$ is in $S_n$ and $I$ is a subset of $\{1,...,n\}$ such that if $i<j$, $w(i)<w(j)$, and $w(j)$ is in $I$, then $w(i)$ is also in $I$. The other side of the correspondence, and the combinatorial algorithm, become suitably complicated. Peter Trapa and I found an exotic analogue of Travkin's correspondence, resulting from the orbits of $Sp(V)$ in $V\times Fl(V)$. I will explain Travkin's results and our analogue.