Two new functors from modules to algebras
type: Item
instance of: scholarly article
title: Two new functors from modules to algebras
zbMATH Open document ID: 0358.15033
DOI: 10.1016/0021-8693(77)90211-3
author: Jacob Towber
published in: Journal of Algebra
publication date: 1977-01-01T00:00:00Z
review text: \textit{G.Higman} [Proc. internat. Conf. Theory Groups, Canberra 1965, 167--173 (1967; Zbl 0242.20040)] has constructed functors of a vectorspace \(E\), related to the irreducible representations of the general linear group; these functors will here be denoted by \(\wedge^\alpha E\) (where \(\alpha\) is a partition). A different construction is given for vectorspaces over fields of characteristic O by \textit{D. B. A. Epstein} [Am. J. Math. 91, 395--414 (1969; Zbl 0201.02501 )], and Epstein's construction has been generalized for \(E\) a module over any commutative ring by \textit{R. W. Carter} and \textit{G.Lusztig} [Math. Z. 136, 193--242 (1974; Zbl 0298.20009)]; this second class of functors will here be denoted by \(\vee_\alpha E\). The present paper concerns the following new constructions and results involving these functors \(\wedge^\alpha E\) and \(\vee_\alpha E: \oplus \vee_\alpha E\) is given the natural structure of a graded anti-commutative associative algebra. (Note that \(\oplus\wedge^\alpha E\) automatically has the structure of a commutative associative algebra, as is implicit in Higman's construction.) Generators and relations are given for the two rings \(\oplus\wedge^\alpha E\), \(\oplus \vee_\alpha E\) which thus make sense for \(E\) a module over a commutative ring. Note 1: In the present paper, \(\wedge^\alpha E\) and \(\vee_\alpha E\) are defined via these generators and relations. The resulting functors of modules \(\wedge^\alpha E\) are naturally isomorphic to those of Higman when \(E\) is a vectorspace, but appear to be new in general. The resulting functors \(\vee_\alpha E\) are naturally Isomorphic to those of Carter-Lusztig when \(E\) is finitely generated projective, but not in general. Note 2: At the end of the paper being reviewed, the author mentioned that these relations for \(\wedge^\alpha E\) may have been already published by Higman and/or his students; Higman has since written the author that this is not the case. If \(\alpha\) …
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 13C99
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 3557974
Wikidata QID: Q29014140
MaRDI profile type: Publication
full work available at URL: 0021-8693(77)90211-3
OpenAlex ID: W1971298218
cites work: Generic free resolutions and a family of generically perfect ideals
cites work: On the modular representations of the general linear and symmetric groups
cites work: On the Foundations of Combinatorial Theory: IX Combinatorial Methods in Invariant Theory
cites work: Group Representations and Functors
cites work: Q5652883
cites work: Q5834477
cites work: Group Characters and the Structure of Groups
cites work: Q5834474
cites work: Q4402068
cites work: Q3284627
cites work: Theory and Application of Plane Partitions. Part 2
cites work: Young symmetry, the flag manifold, and representations of GL(n)
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description: scientific article; zbMATH DE number 3557974