Fusion systems with some sporadic J-components

The research of the first author was partially supported by NSA Young Investigator Grant H98230-14-1-0312 and was supported by an AMS-Simons grant which allowed for travel related to this work.Peer reviewedPostprin

Generalized q,tq,t-Catalan numbers

Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov--Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The $q,t$-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients. For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for $(4,n)$ rational $q,t$-Catalan numbers.Comment: 33 pages; v2: fixed typos and included referee comment

The Gelfand–Graev Representation of U(3,q)

AbstractIn this paper we explicitly calculate the irreducible representations of the endomorphism algebra of the Gelfand–Graev representation of the unitary group U(3,q). In addition, we compute the structure constants of this endomorhphism algebra

Notes on the norm map between the Hecke algebras of the Gelfand–Graev representations of GL(2,q2) and U(2,q)

AbstractLet G˜ be a connected reductive algebraic group defined over the field Fq and let F and F∗ be two Frobenius maps such that Fm=(F∗)m for some integer m. Let G˜F,G˜F∗, and G˜Fm=G˜(F∗)m be the finite groups of fixed points. In this article we consider the case where G˜=GL(2,F¯q), F is the usual Frobenius map so that G˜F=GL(2,q) and F∗ is the twisted Frobenius map such that G˜F∗=U(n,q). In this case, F2=(F∗)2 and G˜F2=G˜(F∗)2=GL(2,q2). This article provides connections between the complex representation theory of these groups using the norm maps (see [C. Curtis, T. Shoji, A norm map for endomorphism algebras of Gelfand–Graev representations, in: Progr. Math., vol. 141, 1997, pp. 185–194]) from the Gelfand–Graev Hecke algebra of GL(2,q2) to the Gelfand–Graev Hecke algebras of both GL(2,q) and U(2,q)