On CRDAHA and finite general linear and unitary groups

Let $${U}_{q} = {U}_{q}({\mathfrak{g}\mathfrak{l}}_{n})$$ be Lusztig’s divided power quantum general linear group over the complex field with parameter q, a root of unity. We investigate the category of finite-dimensional modules over U q . Lusztig’s famous character formula for simple modules over U q is written purely in terms of the affine Weyl group and its Hecke algebra, which are independent ofq. Our result may be viewed as the categorical version of this independence. Moreover, our methods are valid over fields of positive characteristic. The proof uses the modular representation theory of symmetric groups and finite general linear groups, and the notions of sl2-categorification and perverse equivalences.

Weights for symmetric and general linear groups

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups.

Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ∈ cd (G). We determine the graph Δ(G) for the finite simple groups of types A ℓ(q) and 2 A ℓ (q 2), that is, for the simple linear and unitary groups.

Structure Arising from Induction and Jacquet Modules of Representations of Classical p-Adic Groups

On imprimitive representations of finite reductive groups in non-defining characteristic

Asymptotic behaviors in the homology of symmetric group and finite general linear group quandles

We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

This chapter presents a theory of unitary groups over rings parallel to that which Chapter 1 develops for the linear groups. We will introduce the concept of a form ring which will allow for the definition of generalized quadratic forms. The unitary groups — i.e. isometry groups—of these generalized quadratic forms contain as special cases all the usual classical groups. They include in fact the linear groups of Chapter 1. However the proofs of the central theorems of this chapter make important use of the linear results already established. Therefore from a practical point of view, this approach will include all the classical groups, with the exception of the linear groups, under a single umbrella.

Let Γ be a connected reductive group over an algebraically closed field F of positive characteristic and let G be the set of fixed points of Γ under some Frobenius map of Γ. Then G is a finite group of Lie type. The p-modular representations of G for p = char F (the “describing” characteristic case) are closely related to rational representations of Γ, and thus results from the theory of reductive algebraic groups can be used to develop the representation theory of finite groups of Lie type in the describing characteristic case. One of the outstanding open questions in this area is to determine the dimensions of the irreducible representations and the decomposition matrix whose entries record multiplicities of irreducible modules as composition factors of Weyl modules. One way to define Weyl modules is by reducing modulo p (p = char F) the irreducible modules of the complex algebraic group of the same type. Here reduction modulo p happens in both, in the field underlying the group and in the field over which the representing matrices are defined, simultaneously. One of the main tools to study these questions is derived from the Weyl group of Γ. If, for example, Γ is a general linear group, the decomposition matrix of the Weyl modules can be calculated from decomposition numbers of Schur algebras, which in turn can be calculated in terms of symmetric groups. We shall discuss this phenomenon in more detail below.

The degree of the generators of invariant polynomial rings of is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary group---the tensor product of unitary group, and local general linear group---the tensor product of general linear group. For these two groups, we prove polynomial upper bounds on the degree of the generators of invariant polynomial rings. On the other hand, systematic methods are provided to to construct all homogenous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if $r$-copies of them are LU equivalent for some $r$.

By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.

This paper describes parabolic induction, for smooth representations of finite length of the general linear group GL(N, F) of a non-archimedean local field F, in terms of a functor between categories of modules over certain affine Hecke algebras, using the categorical equivalences developed in the Bushnell-Kutzko classification of the admissible dual of GL(N, F). This result is used to prove that the Zelevinsky automorphism, which is an involutive automorphism on the representation ring of the category of smooth representations of finite length of GL(N, F), preserves the irreducible representations. The result also implies a simplification of the study of multiplicities of composition factors of induced representations. 1991 Mathematics Subject Classification: 11S37, 22E50.