Algebra Seminar: Henderson -- Mirabolic and exotic Robinson-Schensted correspondences

The Algebra Seminar on Friday 13 May will be given by Anthony Henderson.  

Thank you to Anthony Henderson and Andrew Mathas for running the seminar during my
absence.  

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Speaker: Anthony Henderson (University of Sydney) 

Date: Friday 13 May

Time: 12.05-12.55pm 

Venue: Carslaw 175 

Title: 

Mirabolic and exotic Robinson-Schensted correspondences 

Abstract: 

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) x 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 x Fl(V) x 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 x Fl(V).  I will explain Travkin’s results and our analogue.

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Anne Thomas - [email protected] Seminar website -
http://www.maths.usyd.edu.au/u/AlgebraSeminar/