New York Journal of Mathematics
Volume 23 (2017) 165-191


Jason Fulman, Rita Jiménez Rolland, and Jennifer C. H. Wilson

Generating functions and statistics on spaces of maximal tori in classical Lie groups

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Published: February 8, 2017
Keywords: Generating functions, finite groups of Lie type, maximal tori, symmetric group, hyperoctahedral group, signed permutation group, representation stability, homological stability
Subject: 57T15, 05A15, 20G40, 20G05

In this paper we use generating function methods to obtain new asymptotic results about spaces of F-stable maximal tori in GLn(\overline{Fq}), Sp2n(\overline{Fq}), and SO2n+1(\overline{Fq}). We recover stability results of Church-Ellenberg-Farb and Jiménez Rolland-Wilson for "polynomial'' statistics on these spaces, and we compute explicit formulas for their stable values. We derive a double generating function for the characters of the cohomology of flag varieties in type B/C, which we use to obtain analogs in type B/C of results of Chen: we recover "twisted homological stability'' for the spaces of maximal tori in Sp2n(C) and SO2n+1(C), and we compute a generating function for their "stable twisted Betti numbers''. We also give a new proof of a result of Lehrer using symmetric function theory.


Fulman was partially supported by Simons Foundation grant 400528.
Jiménez Rolland is grateful for the financial support from PAPIIT-UNAM grant IA100816.

Author information

Jason Fulman:
Department of Mathematics, University of Southern California, Los Angeles, CA, United States 90089-2532

Rita Jiménez Rolland:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Oaxaca de Juárez, Oaxaca, México 68000

Jennifer C. H. Wilson:
Department of Mathematics, Stanford University, Stanford, CA, United States 94305