A supercharacter theory for involutive algebra groups

Simple classical groups of Lie type are determined by their character degrees

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Classical groups over finite fields, including symplectic groups, unitary groups and orthogonal groups, are finite simple groups of crucial importance. The structure and order of classical groups are the basic problems in group theory. Geometry of classical groups has been found extremely useful in many fields, such as combinatorial mathematics, coding and cryptographic theory. In this paper, a class of quasi-classical groups are introduced, which partially generalize the concepts of symplectic groups and orthogonal groups. We determine the structure and order of quasi-classical groups. As an application, we discuss the relationship between quasi-classical groups and self-dual normal bases over finite fields.

A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and double-contour integral formulas for the q-specialized characters.

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

Supercharacter theories of semiextraspecial p-groups and Frobenius groups

A supercharacter analogue for normality

In this paper we deduce a graded version of Quillen–Suslin’s Local-Global Principle for the traditional classical groups, viz. general linear, symplectic and orthogonal groups and establish its equivalence of the normality property of the respective elementary subgroups. This generalizes previous result of Basu–Rao–Khanna. Then, as an application, we establish an analogue Local-Global Principle for the commutator subgroups of the special linear and symplectic groups. Finally, by using Swan–Weibel’s homotopy trick, we establish graded analogue of the Local-Global Principle for the transvection subgroups of the full automorphism groups of the linear, symplectic and orthogonal modules. This generalizes the previous result of Bak–Basu–Rao.

A supercharacter theory for a finite group [Formula: see text] is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility conditions. The aim of this paper is to give a description of some finite simple groups with exactly three or four supercharacter theories.

\textbfAbstract. We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic matrices and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table. \bigbreak

ABSTRACTIn this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are ℤ3 and S3. We also show that the only nonsolvable group with two supercharacter theories is Sp(6,2).

The spectrum of a finite group G is the set of the orders of elements of G. In the present article, we obtain an arithmetical description of the spectra of finite symplectic and orthogonal groups. In particular, we describe the spectra of finite simple groups in these two classes.

We classify the irreducible complex characters of the symplectic groups Sp 2n (q) and the orthogonal groups , Spin 2n+1(q) of degrees up to the bound D, where D = (q n − 1)q 4n−10/2 for symplectic groups, D = q 4n−8 for orthogonal groups in odd dimension, and D = q 4n−10 for orthogonal groups in even dimension.

Let A be a linear transformation with minimum polynomial mA(t) and characteristic polynomial cA(t). We say that A is cyclic if mA(t) = cA(t), semisimple if mA(t) has no repeated factors, and separable if it is both cyclic and semisimple. The proportions of elements of GL(d, q) which are cyclic or separable have been investigated by, amongst others, Neumann and Praeger [7], Wall [10], and Fulman [4] who also studies semisimple elements. Making use of their results, we gave estimates in [1] for these proportions in SL(d, q). Fulman, Neumann and Praeger [5] have recently found good estimates for the corresponding proportions in the finite classical groups, U(d, q2), Sp(2m, q) and O±(d, q). They raise the question of whether similar methods may be used to calculate the proportions in various subgroups and supergroups of the classical groups. In a series of three papers, we shall show how this may be done. In this, the first paper, we deal with the special unitary groups SU(d, q2), consisting of unitary transformations with determinant 1. In the second paper [2] we shall focus on the conformal groups GSp(2m, q) and GO±(d, q), and in the third paper [3] we shall consider how the method may be extended to a number of groups of orthogonal type, including the special orthogonal group and the derived group of the orthogonal group. Our method in [1] involved the vector space cycle index of Kung [6] and Stong [8], which we describe briefly in Section 2. Similarly, in the current series