Super-characters of finite unipotent groups of types Bn, Cn and Dn (original) (raw)

Basic characters of the unitriangular group (for arbitrary primes)

PROCEEDINGS-AMERICAN MATHEMATICAL …, 2002

Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elements. In this paper we consider the basic (complex) characters of Un(q) and we prove that every irreducible (complex) character of Un(q) is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption p ≥ n, where p is the characteristic of the field Fq.

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

Journal of Group Theory, 2000

Let GL(n, Fq) τ and U(n, F q 2 ) τ denote the finite general linear and unitary groups extended by the transpose inverse automorphism, respectively, where q is a power of 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(n, Fq) or U(n, F q 2 ). Let yτ be an element of GL(n, Fq) τ or U(n, F q 2 ) τ such that (yτ ) 2 is regular unipotent in GL(n, Fq) or U(n, F q 2 ), respectively. We show that χ(yτ ) = ±1 if χ(1) is prime to p and χ(yτ ) = 0 otherwise. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.

Counting characters of small degree in upper unitriangular groups

Journal of Pure and Applied Algebra, 2011

Let U n denote the group of upper n × n unitriangular matrices over a fixed finite field F of order q. That is, U n consists of upper triangular n × n matrices having every diagonal entry equal to 1. It is known that the degrees of all irreducible complex characters of U n are powers of q. It was conjectured by Lehrer that the number of irreducible characters of U n of degree q e is an integer polynomial in q depending only on e and n. We show that there exist recursive (for n) formulas that this number satisfies when e is one of 1, 2 and 3, and thus show that the conjecture is true in those cases.

Supercharacters of the sylow p-subgroups of the finite symplectic and orthogonal groups

Pacific Journal of Mathematics, 2009

We define and study supercharacters of the classical finite unipotent groups of types B n (q), C n (q) and D n (q). We show that the results we proved in 2006 remain valid over any finite field of odd characteristic. In particular, we show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. In addition, we prove that the unitary vector space spanned by all the supercharacters is closed under multiplication, and establish a formula for the supercharacter values. As a consequence, we obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we give a combinatorial description of all the irreducible characters of maximum degree in terms of the root system, by showing how they can be obtained as constituents of particular supercharacters.

Groups where all the irreducible characters are super-monomial

arXiv (Cornell University), 2008

Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if G is an M-group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if G is an M-group of odd order where every irreducible character is a {p}-lift for some prime p. We say that a group where irreducible character is super monomial is a super M-group. We use our results to find an example of a super M-group that has a subgroup that is not a super M-group. MSC primary : 20C15

On character values in finite groups

Bulletin of the Australian Mathematical Society, 1977

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) -X(u) = c or X(u) = 0 . It is shown that c is an even positive integer and all such groups with a -8 are described.

On characters in the principal 2-block, II

Journal of the Australian Mathematical Society, 1979

Let k be a non-zero complex number and let u and v be elements of a finite group G. Suppose that at most one of u and v belongs to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(v) -X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G, if and only if G/O(G) is isomorphic to one of the following groups: C t , PSL(2, 2") or PSi(2, 5 2o+1 ), where n>2 and a> 1.