Blocks in flat families of finite-dimensional algebras

Abstract

We study the behavior of blocks in flat families of finite-dimensional algebras. In a general setting we construct a finite directed graph encoding a stratification of the base scheme according to the block structures of the fibers. This graph can be explicitly obtained when the central characters of simple modules of the generic fiber are known. We show that the block structure of an arbitrary fiber is completely determined by "atomic" block structures living on the components of a Weil divisor. As a byproduct, we deduce that the number of blocks of fibers defines a lower semicontinuous function on the base scheme. We furthermore discuss how to obtain information about the simple modules in the blocks by generalizing and establishing several properties of decomposition matrices by Geck and Rouquier.

Computational aspects of Calogero-Moser spaces

“…χ ′ for some χ, χ ′ } be the locus of all C where the C-families are not equal to (and thus coarser than) the generic families. It follows from [Thi2,Theorem 1.3] that this locus is a closed subset of Spec(C[ C]) which is either empty or pure of codimension 1. It is obviously contained in…”

“…, H r be the Calogero-Moser hyperplanes and for each j let C j be the corresponding prime ideal in C[ C]. It follows from [Thi2,Lemma 3.6] that for c ∈ C CM we have (5.5)…”

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.…”

“…Then one can define the restricted rational Cherednik algebra also over the rational function field C(C). 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…”

“…For decompositiongeneric parameters, the matrices in Definition 2.5 do not depend on the particular parameter by construction of the decomposition map. It is shown in[Thi16;Thi18] that both types of generic parameters form a non-empty Zariski open subset of the parmater space and that block-generic parameters are decomposition-generic. Moreover, it follows from[BST18] that the set of block-generic parameters is the complement of a finite hyperplane arrangement.…”