# 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.