A sum rule for scale factors in the Goldie rank polynomials
type: Item
instance of: scholarly article
title: A sum rule for scale factors in the Goldie rank polynomials
zbMATH Open document ID: 0699.17014
DOI: 10.1016/0021-8693(88)90022-1
author: Anthony Joseph
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The author by applying the results of a book of \textit{G. Lusztig} [Characters of reductive groups over a finite field. Ann. Math. Stud. 107 (1984; Zbl 0556.20033] to that of his own paper [Lect. Notes Math. 1243, 144-188 (1987; Zbl 0621.17006)] proves that (i) if \(\tau_ 1\), \(\tau_ 2\) are involutions in a Lusztig cell then Soc \(L(L(\tau_ 1\lambda),L(\tau_ 2\lambda))\) is determined by multiplying appropriate irreducible characters of A and identifying coefficients. In particular, if \(A=S_ 5\) then it can happen that \(L(L(\tau\lambda),L(\tau\lambda)\)) is not multiplicity free; (ii) if \(\tau\) is an involution in a Lusztig cell then \(z_{\tau}\) is the degree of an irreducible character of A; (iii) \(z_{\omega}\) divides \(| A|\) for every \(\omega\) of a Lusztig cell; (iv) if \(A_{{\mathcal D}{\mathcal C}}\) is commutative then for each left cell \({\mathcal C}\) of \({\mathcal D}{\mathcal C}\) the modules \(\{Soc\;L(L(\omega\lambda),L(\omega\lambda)):\) \(\omega\in {\mathcal C}\}\) generate the same subgroup of the Grothendieck group as the modules \(\{L(M(\lambda),L(\tau\lambda)):\) \(\tau\in {\mathcal C}\cap \Sigma\}.\) See also the following review.
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 4146210
zbMATH Keywords: semisimple Lie algebra
zbMATH Keywords: Kazhdan-Lusztig
zbMATH Keywords: polynomials
zbMATH Keywords: component group
zbMATH Keywords: nilpotent orbit
zbMATH Keywords: ratios of Goldie
zbMATH Keywords: ranks
zbMATH Keywords: Weyl group
zbMATH Keywords: Hecke algebra module
zbMATH Keywords: involutions
zbMATH Keywords: Lusztig cell
zbMATH Keywords: irreducible characters
reviewed by: Yu. N. Maltsev
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90022-1
OpenAlex ID: W2021085531
cites work: Q5340151
cites work: The modular representation algebra of a finite group
cites work: Q5530497
cites work: Q3855349
cites work: Goldie rank in the enveloping algebra of a semisimple Lie algebra. III
cites work: On the variety of a highest weight module
cites work: Q3667037
cites work: Q3758092
cites work: Multiplicity of the Adjoint Representation in Simple Quotients of the Enveloping Algebra of a Simple Lie Algebra
cites work: Modules of t-Finite Vectors Over Semi-Simple Lie Algebras
cites work: On a theorem of Benson and Curtis
cites work: Q4750022
cites work: Characters of Reductive Groups over a Finite Field. (AM-107)
cites work: Q4723949
cites work: Q3686887
cites work: Q3710659
cites work: Second Commutant Theorems in Enveloping Algebras
cites work: Q4145843
description: scientific article; zbMATH DE number 4146210