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.