Multivariate Rogers-Szeg\"o polynomials and flags in finite vector spaces

We give a recursion for the multivariate Rogers-Szeg\"o polynomials, along with another recursive functional equation, and apply them to compute special values. We also consider the sum of all $q$-multinomial coefficients of some fixed degree and length, and give a recursion for this sum which follows from the recursion of the multivariate Rogers-Szeg\"o polynomials, and generalizes the recursion for the Galois numbers. The sum of all $q$-multinomial coefficients of degree $n$ and length $m$ is the number of flags of length $m-1$ of subspaces of an $n$-dimensional vector space over a field with $q$ elements. We give a combinatorial proof of the recursion for this sum of $q$-multinomial coefficients in terms of finite vector spaces

Totally orthogonal finite simple groups

We prove that if $G$ is a finite simple group, then all irreducible complex representations of $G$ by be realized over the real numbers if and only if every element of $G$ may be written as a product of two involutions in $G$. This follows from our result that if $q$ is a power of $2$, then all irreducible complex representations of the orthogonal groups $\mathrm{O}^{\pm}(2n, \mathbb{F}_q)$ may be realized over the real numbers. We also obtain generating functions for the sums of degrees of several sets of unipotent characters of finite orthogonal groups, and we obtain a twisted version of our main result for a broad family of finite classical groups

Alvis-Curtis duality, central characters, and real-valued characters

We prove that Alvis-Curtis duality preserves the Frobenius-Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius-Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius-Schur indicators of certain real-valued, irreducible, Frobenius-invariant Deligne-Lusztig characters, and the Frobenius-Schur indicators of real-valued regular and semisimple characters of finite unitary groups

Real representations of finite symplectic groups over fields of characteristic two

We prove that when $q$ is a power of $2$, every complex irreducible representation of $\mathrm{Sp}(2n, \mathbb{F}_q)$ may be defined over the real numbers, that is, all Frobenius-Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of $\mathrm{Sp}(2n, \mathbb{F}_q)$, or of $\mathrm{SO}(2n+1, \mathbb{F}_q)$, for any prime power $q$

On the number of real classes in the finite projective linear and unitary groups

We show that for any $n$ and $q$, the number of real conjugacy classes in $\mathrm{PGL}(n, \mathbb{F}_q)$ is equal to the number of real conjugacy classes of $\mathrm{GL}(n, \mathbb{F}_q)$ which are contained in $\mathrm{SL}(n, \mathbb{F}_q)$, refining a result of Lehrer, and extending the result of Gill and Singh that this holds when $n$ is odd or $q$ is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $\mathrm{PGU}(n, \mathbb{F}_q)$, and equal to the number of real conjugacy classes of $\mathrm{U}(n, \mathbb{F}_q)$ which are contained in $\mathrm{SU}(n, \mathbb{F}_q)$, refining results of Gow and Macdonald. We also give a generating function for this common quantity

A factorization result for classical and similitude groups

For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of M{\oe}glin, Vign\'{e}ras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representations to its dual

Gelfand-Graev characters of the finite unitary groups

Gelfand-Graev characters and their degenerate counterparts have an important role in the representation theory of finite groups of Lie type. Using a characteristic map to translate the character theory of the finite unitary groups into the language of symmetric functions, we study degenerate Gelfand-Graev characters of the finite unitary group from a combinatorial point of view. In particular, we give the values of Gelfand-Graev characters at arbitrary elements, recover the decomposition multiplicities of degenerate Gelfand-Graev characters in terms of tableau combinatorics, and conclude with some multiplicity consequences

A computational approach to the Frobenius-Schur indicators of finite exceptional groups

We prove that the finite exceptional groups $F_4(q)$, $E_7(q)_{\mathrm{ad}}$, and $E_8(q)$ have no irreducible complex characters with Frobenius-Schur indicator $-1$, and we list exactly which irreducible characters of these groups are not real-valued. We also give an exact list of complex irreducible characters of the Ree groups ${^2 F_4}(q^2)$ which are not real-valued, and we show the only character of this group which has Frobenius-Schur indicator $-1$ is the cuspidal unipotent character $\chi_{21}$ found by M. Geck.Comment: Version 2 has some corrections to Table A.1 for the case q=2, added exceptional cases to Lemmas 5.1 and 5.2, and updated Section

A product formula for multivariate Rogers-Szeg\"o polynomials

Let $H_n(t)$ denote the classical Rogers-Szeg\"o polynomial, and let \tH_n(t_1, \ldots, t_l) denote the homogeneous Rogers-Szeg\"o polynomial in $l$ variables, with indeterminate $q$. There is a classical product formula for $H_k(t)H_n(t)$ as a sum of Rogers-Szeg\"o polynomials with coefficients being polynomials in $q$. We generalize this to a product formula for the multivariate homogeneous polynomials \tH_n(t_1, \ldots, t_l). The coefficients given in the product formula are polynomials in $q$ which are defined recursively, and we find closed formulas for several interesting cases. We then reinterpret the product formula in terms of symmetric function theory, where these coefficients become structure constants

On the characteristic map of finite unitary groups

In his classic book on symmetric functions, Macdonald describes a remarkable result by Green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions. This connection allows us to analyze the representation theory of the general linear group via symmetric group combinatorics. Using the work of Ennola, Kawanaka, Lusztig and Srinivasan, this paper describes the analogous setting for the finite unitary group. In particular, we explain the connection between Deligne-Lusztig theory and Ennola's efforts to generalize Green's work, and deduce various representation theoretic results from these results. Applications include finding certain sums of character degrees, and a model of Deligne-Lusztig type for the finite unitary group, which parallels results of Klyachko and Inglis and Saxl for the finite general linear group