On Schur algebras and little Schur algebras
type: Item
instance of: scholarly article
title: On Schur algebras and little Schur algebras
zbMATH Open document ID: 1241.17016
author: Qiang Fu
published in: Journal of Algebra
publication date: 2009-10-13T00:00:00Z
review text: Let \(\eta\) be a possibly infinite consecutive segment in \({\mathbb Z}\). The quantized enveloping algebra \(U(\eta)\) of \({\mathfrak gl}(\eta)\) was realized in [\textit{A. A. Bejlinson, G. Lusztig} and \textit{R. MacPherson}, ``A geometric setting for the quantum deformation of \(GL_n\)'', Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)] using a geometric construction of the \(q\)-Schur algebras \(S_Q(\eta,r)\). An algebra homomorphism \(\xi_r: U(\eta)\to S_Q(\eta,r)\) was constructed. This map is surjective when \(\eta\) is finite. Poincaré-Birkhoff-Witt, Beilinson-Lusztig-MacPherson, and monomial bases have been constructed for its image \(U(\eta,r)\) by different authors. The present paper addresses the problem of finding a BLM-realization for the classical enveloping algebra \(U(gl_\eta)\) over the field of rational numbers and for the restricted enveloping algebra \(u_k(gl_\eta)\) over the field with \(p\) elements, making use of classical and little Schur algebras. This result allows the construction of bases for the classical analogue of \(U(\eta,r)\) and for the so-called little Schur algebras \(u_k(\eta,r)\). Such bases are also used to obtain the following result. Let \(A={\mathbb Z}[\varepsilon^{\pm1}]\) for \(\varepsilon\) a primitive \(p\)-th root of \(1\) and let \(u_A(\eta)\) the infinitesimal quantum group of \(gl_\eta\). In analogy to the fact that the specialization of \(u_A(\eta)\) at \(\varepsilon=1\) is isomorphic to \(u_k(\eta)\), the authors provide an \(A\)-form for little \(q\)-Schur algebras and show that their specializations at \(\varepsilon=1\) are isomorphic to the corresponding little Schur algebras.
reviewed by: G. Carnovale
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 20G43
zbMATH DE Number: 5614724
zbMATH Keywords: Schur algebras
zbMATH Keywords: little Schur algebras
zbMATH Keywords: little \(q\)-Schur algebras
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.04.027
OpenAlex ID: W2063499139
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DOI: 10.1016/J.JALGEBRA.2009.04.027
description: scientific article; zbMATH DE number 5614724