Brauer-type reciprocity for a class of graded associative algebras

Abstract. In this paper, we introduce baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter t = 0 (the so-called rational Cherednik algebras at parameter t = 0) and present their most basic properties. As an example, we use baby Verma modules to answer several problems posed by Etingof and Ginzburg, [5], and give an elementary proof of a theorem of Finkelberg and Ginzburg, [6].

show abstract

“…4.3. The following proposition gathers results from [14,Section 3]. Recall that (v) Let P (S) be the projective cover of L(S).…”

Section: 2mentioning

confidence: 86%

Baby Verma Modules for Rational Cherednik Algebras

Gordon¹

2003

Bull. London Math. Soc.

165

View full text Add to dashboard Cite

show abstract

“…The results of Holmes and Nakano [9] apply in our setup, since all the simple modules L χ (λ) are of type M. In particular, by [9, Thms. 4.5 and 5.1] the projective cover P χ (λ) of L χ (λ) has a baby Verma filtration, and for any λ, µ ∈ Λ χ one has the Brauer type reciprocity (P χ (λ) : Z χ (µ)) = [Z χ (µ) : L χ (λ)], where (P χ (λ) : Z χ (µ)) is the multiplicity of Z χ (µ) appearing in the baby Verma filtration of P χ (λ), and [Z χ (µ) : L χ (λ)] is the multiplicity of L χ (λ) in a composition series of Z χ (µ).…”

Section: Structure Of U χ (G) For Semisimple χmentioning

confidence: 93%

Representations of Lie superalgebras in prime characteristic II: The queer series

Wang

Zhao

2011

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

a b s t r a c tThe modular representation theory of the queer Lie superalgebra q(n) over characteristic p > 2 is developed. We obtain a criterion for the irreducibility of baby Verma modules with semisimple p-characters χ and a criterion for the semisimplicity of the corresponding reduced enveloping algebras U χ (q(n)). A (2p)-power divisibility of dimensions of q(n)-modules with nilpotent p-characters is established. The representation theory of q(2) is treated in detail. We formulate a Morita super-equivalence conjecture for q(n) with general p-characters which is verified for n = 2.

show abstract

“…As we see below, this number is independent of the choice of filtration. Applying standard arguments from, for example, [5,15], we have the following results about the category F.…”

Section: Filtrations By Kac Supermodulesmentioning

confidence: 94%

Cohomology and support varieties for Lie superalgebras II

Boe

Kujawa

Nakano

2008

Proceedings of the London Mathematical Society

View full text Add to dashboard Cite

In [2] (Preprint, 2006, arXiv:math.RT/0609363) the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra, and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties of Kac supermodules for Type-I Lie superalgebras and of the simple supermodules for gl(m|n). The latter result verifies our earlier conjecture for gl(m|n). In our investigation we also delineate several of the major differences between Type-I versus Type-II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.

show abstract

Brauer-type reciprocity for a class of graded associative algebras

Cited by 46 publications

References 8 publications

Baby Verma Modules for Rational Cherednik Algebras

Baby Verma Modules for Rational Cherednik Algebras

Representations of Lie superalgebras in prime characteristic II: The queer series

Cohomology and support varieties for Lie superalgebras II

Contact Info

Product

Resources

About