On involutions in symmetric groups and a conjecture of Lusztig

2015/07/03 by Jun Hu, Jing Zhang, Hu, Jun +1
Mathematics · #20C08 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:20C08

paper · pdf · doi:10.48550/arxiv.1507.00872

arxiv created 2015/07/03 · arxiv updated 2015/07/06

Abstract

Let (W, S) be a Coxeter system equipped with a fixed automorphism ∗ of order ≤ 2 which preserves S. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of "twisted" involutions was naturally endowed with a module structure of the Hecke algebra of (W, S). Lusztig has conjectured that this module is isomorphic to the right ideal of the Hecke algebra (with Hecke parameter u²) associated to (W,S) generated by the element X_∅:=∑_w^∗=wu^-ℓ(w)T_w. In this paper we prove this conjecture in the case when ∗=id and W is the symmetric group on n letters.

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