C. Ryan Vinroot

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents

Abstract Let GL and U denote the finite general linear and unitary groups extended by the transpose inverse automorphism, respectively, where q is a power of the prime p. Let n be odd, and let χ be an irreducible character of either of these groups which is an extension of a real-valued character of GL or U. Let yτ be an element of GL or U such that (yτ)2 is regular unipotent in GL or U, respectively. We show that is prime to p and otherwise. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.

A Factorization in GSp(V)

Let V be a vector space over the field F such that char(F) ≠ 2, and let V have a skew-symmetric nondegenerate bilinear form. Wonenburger proved that any element g of Sp(V) is the product of two skew-symplectic involutions. Let GSp(V) be the group of general similitudes with similitude character μ. We give a generalization of Wonenburgers result in the following form. Let g ∈ GSp(V) with μ (g) = β. Then g = t 1 t 2 such that t 1 is a skew-symplectic involution, and t 2 is such that and μ (t 2) = −β. One application that follows from this result is a necessary and sufficient condition for an element of GL(V) to be conjugate to a scalar multiple of its inverse. Another result is that we find an extension of the group , for q ≡ 3(mod 4), all of whose complex representations have real-valued characters.Let V be a vector space over the field F such that char(F) ≠ 2, and let V have a skew-symmetric nondegenerate bilinear form. Wonenburger proved that any element g of Sp(V) is the product of two skew-symplectic involutions. Let GSp(V) be the group of general similitudes with similitude character μ. We give a generalization of Wonenburgers result in the following form. Let g ∈ GSp(V) with μ (g) = β. Then g = t 1 t 2 such that t 1 is a skew-symplectic involution, and t 2 is such that and μ (t 2) = −β. One application that follows from this result is a necessary and sufficient condition for an element of GL(V) to be conjugate to a scalar multiple of its inverse. Another result is that we find an extension of the group , for q ≡ 3(mod 4), all of whose complex representations have real-valued characters.

Involution models of finite Coxeter groups

Abstract Let G be a finite Coxeter group. Using previous results on Weyl groups, and covering the cases of non-crystallographic groups, we show that G has an involution model if and only if all of its irreducible factors are of type A n , B n , D 2n+1, H 3, or I 2(n).

Duality, central characters, and real-valued characters of finite groups of Lie type

We prove that the duality operator 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 char acter. 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 u nitary groups. Given a finite group G, and an irreducible finite dimensional complex representation ( , V ) of G with character , it is a natural question to ask what smallest field extension of is necessary to define a matrix representation correspondin g to ( , V ). If ( ) is the smallest extension of containing the values of , then the Schur index of over may be defined to be the smallest degree of an extension of ( ) over which ( , V ) may be defined. One may also consider the Schur index of over , which is 1 if ( , V ) may be defined over ( ), and 2 if it is not. If is a real-valued character, then the Schur index of over indicates whether ( , V ) may be defined over the real numbers. The Brauer‐Speiser Theorem states that if is a real-valued character, then the Schur index of over is either 1 or 2, and if the Schur index of over is 2, then the Schur index of over is 2. As finite groups of Lie type are of fundamental importance in t he theory of finite groups, it is of interest to understand the Schur indices ove r of their complex representations. In work of Gow and Ohmori [10, 11, 12, 19, 20], the Schur indices over of many irreducible characters of finite classical groups ar e determined. For the characters which are not covered by methods of Gow and Ohmori, it seems that the computation of the Schur index over is significantly more difficult. One example is that of the special linear group, which was completed by Turull [24]. Through methods developed by Lusztig, Geck, and Ohmori, the Schur index of unipotent characters have

Real classes of finite special unitary groups

Abstract We classify all real and strongly real classes of the finite special unitary group SU n ⁢ ( q )

A weight statistic and partial order on products of m-cycles

{\mathrm{SU}_{n}(q)}

Strongly real classes in finite unitary groups of odd characteristic

. Unless q ≡ 3 ⁢ ( mod ⁡ 4 )

STRONG REALITY PROPERTIES OF NORMALIZERS OF PARABOLIC SUBGROUPS IN FINITE COXETER GROUPS

{q\equiv 3\;(\operatorname{mod}4)}

A note on orthogonal similitude groups

and n | 4