Two new functors from modules to algebras
\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 \(\alpha\) and \(\alpha\ast\) are dual partitions, a natural pairing \(\wedge^\alpha E^\oplus \vee_{\alpha\ast} E^\ast \to k\) is constructed, which exhibits \(\wedge^\alpha E\) and \(\vee_{\alpha\ast} E^\ast\) a dual modules if \(E\) is finitely generated projective. If \(E\) is free of rank \(r\), then \(\wedge^\alpha E\), \(\vee_\alpha E\) are free of rank depending only on \(\alpha\) and \(r\) (this is proved by Carter and Lusztig for \(\vee_\alpha E\)). If \(k\) is a field of characteristic 0, an explicit isomorphism is constructed between \(\wedge^\alpha E\) and \(\vee_{\alpha^\ast} E\); this yields new structure on the affine coordinate ring of the flag manifold (in characteristic 0 only), namely a natural graded anti-commutative associative wedge-product operation. One final concept in this paper which appears to be new is that of Young symmetry, with respect to a partition of \(n\), of a function of \(n\) variables taking values in a \(\mathbb{Z}\)-module; this coincides with Young's original concept if the function takes values in \(\mathbb{Q}\)-module. In this connection it should be mentioned that the citation on p. 100 for a certain combinatorial lemma, now appears not to exist in the literature; the author will be glad to mail a proof of this lemma on request.
scientific article; zbMATH DE number 3557974