Supercharacters of the sylow p-subgroups of the finite symplectic and orthogonal groups (original) (raw)
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A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups
Journal of Algebra, 2009
We define the superclasses for a classical finite unipotent group U of type B n (q), C n (q), or D n (q), and show that, together with the supercharacters defined in [6], they form a supercharacter theory in the sense of . In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As as consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of U , and prove various orthogonality relations for supercharacters (similar to the well-known orthogonality relations for irreducible characters).
Super-characters of finite unipotent groups of types Bn, Cn and Dn
Journal of Algebra, 2006
We define and study super-characters (over the complex field) of the classical finite unipotent groups of types B n , C n and D n. Under the assumption that the prime is sufficiently large, we extend the known results for the unitriangular group proved by the first author in the papers: [C.A.M. André, Basic characters of the unitriangular group, J. Algebra 175 (1995) 287-319], and [C.A.M. André, Basic characters of the unitriangular group (for arbitrary primes), Proc. Amer. Math. Soc. 130 (7) (2002) 1943-1954]. In particular, we prove that every irreducible (complex) character occurs as a constituent of a unique super-character. We also give a combinatorial description of all the irreducible characters of maximum degree.
Supercharacters of discrete algebra groups
2021
The concept of a supercharacter theory of a finite group was introduced by Diaconis and Isaacs in [15] as an alternative to the usual irreducible character theory, and exemplified with a particular construction in the case of finite algebra groups. We extend this construction to arbitrary countable discrete algebra groups, where superclasses and indecomposable supercharacters play the role of conjugacy classes and indecomposable characters, respectively. Our construction can be understood as a cruder version of Kirillov’s orbit method and a generalisation of Diaconis and Isaacs construction for finite algebra groups. However, we adopt an ergodic theoretical point of view. The theory is then illustrated with the characterisation of the standard supercharacters of the group of upper unitriangular matrices over an algebraic closed field of prime characteristic.
A supercharacter theory for involutive algebra groups
Journal of Algebra, 2015
If JJ is a finite-dimensional nilpotent algebra over a finite field kk, the algebra group P=1+JP=1+J admits a (standard) supercharacter theory as defined in [16]. If JJ is endowed with an involution σ, then σ naturally defines a group automorphism of P=1+JP=1+J, and we may consider the fixed point subgroup CP(σ)={x∈P:σ(x)=x−1}CP(σ)={x∈P:σ(x)=x−1}. Assuming that kk has odd characteristic p, we use the standard supercharacter theory for P to construct a supercharacter theory for CP(σ)CP(σ). In particular, we obtain a supercharacter theory for the Sylow p-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by André and Neto in and for the special case of the symplectic and orthogonal groups.
Supercharacters and pattern subgroups in the upper triangular groups
2010
Let Un(q) denote the upper triangular group of degree n over the finite field Fq with q elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters Irr(Un(q)). In this paper we present a correspondence between supercharacters and pattern subgroups of the form U k (q)∩ w U k (q) where w is a monomial matrix in GL k (q) for some k < n.
Real representations of the finite orthogonal and symplectic groups of odd characteristic
Journal of Algebra, 1985
Let q be a power of an odd prime p. Let Sp(2n, q) denote the symplectic group of degree 2n over GF(q). Let O+(n, q) denote the split orthogonal group of degree n over GF(q), corresponding to a symmetric form of maximal Witt index, and 0 -(n, q) denote the non-split orthogonal group. Unless we need to be more specific, we will simply write O(n, q) for either of these two orthogonal groups. In addition, we will often abbreviate these symbols further to O(n) and Sp(2n) whenever q is understood to be fixed. It is known from conjugacy class considerations that all complex irreducible characters of the finite orthogonal groups O(n) are real-valued. The same is true of the symplectic groups Sp(2n, q) provided that q z 1 (mod 4). When q= 3 (mod 4), not all characters of Sp(2n, q) are realvalued. However, as the total number of irreducible characters is a manic polynomial in q of degree n and this is true of the number of real-valued irreducible characters, we can assert that the majority of characters is realvalued in this case. It is of interest to know whether or not the real-valued characters of these families of groups are the characters of representations that are defined over the real numbers. This paper gives a solution to this problem by proving the following theorem. THEOREM 1. Let q be a power of an odd prime. Each complex irreducible character of O(n, q) is the character of a real representation. Each nonfaithful real-valued irreducible character of Sp(2n, q) is the character of a real representation, whereas each faithful real-valued irreducible character of the group has Schur index 2 over the real numbers.
2010
The standard supercharacter theory of the finite unipotent upper-triangular matrices U n (q) gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of U m (q) ⊆ U n (q) for m ≤ n lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of U n (q) is a nonnegative integer linear combination of supercharacters of U m (q) (in fact, it is polynomial in q). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of U n (q), this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs.