Decomposition Matrices are Generically Trivial

Abstract

ABSTRACT. We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier of an algebra which is free of finite dimension over a noetherian integral domain and which splits over the fraction field of this ring are generically trivial, i.e., they are trivial in an open neighborhood of the generic point of the spectrum of the base ring. This generalizes a classical result by Brauer in modular representation theory of finite groups. We furthermore show that this is true precisely on an open set in case all fibers of the algebra split. In this way we get a stratification of the base scheme such that decomposition maps are trivial on each stratum. Moreover, this defines a new discriminant of such algebras which generalizes Schur elements of simple modules for symmetric split semisimple algebras. We provide some extensions to the theory of decomposition maps allowing us to work without the usual normality assumption on the base ring.

Blocks in flat families of finite-dimensional algebras

“…Let := + , where := Rad( ) ∩ O . The arguments in [42] show that is an O -lattice of + Rad( ) and that the reduction in the maximal ideal m of O is equal to O (m) + Rad( O (m)). We thus have dim ( + Rad( )) = dim k(m) ( O (m) + Rad( O (m))).…”

Restricted rational Cherednik algebras

“…As an example, we can take c = • to be the generic point defined by the zero ideal in which case we have H| • = H. From now on, we assume that C/c is normal. 13 The theory of decomposition maps by Geck and Rouquier [35] (see also [36] and [74]) shows that for any c ∈ V(c) there is a morphism d c…”

“…We then have a unique decomposition map d p A : G 0 (A K ) → G 0 (A(p)) as defined by Geck and Rouquier [35], see also [74]. We recall from [74] that d p A can be realized by a discrete valuation ring in K with maximal ideal lying above p. The morphism ρ| Ap describes an A(p)-module S and since it is surjective, the module S must be simple. Furthermore, the morphism ρ| Ap describes an R p -free A p -form S of S. We thus have d p A ([S]) = [S], the class of a simple module.…”

Wreath Macdonald polynomials at q=t as characters of rational Cherednik algebras

“…Thi17, Sec. 3] a notion of generic parameters was introduced which relies on the general theory in[Thi16;Thi18]. Intuitively, the representation theory of H c is "the same" for all generic c. To make this more precise, let C := (C s ) s∈S be a set of indeterminates over C such that C s = C t whenever s and t are conjugate.…”

“…Let us denote this (generic) algebra by H. It follows from[Thi18] that the blocks of H c are unions of blocks of H. We thus call c block-generic if the blocks coincide. On the other hand, it follows from[Thi16] that there is a map, called decomposition map, from the (graded) Grothendieck group of H to the one of H…”