On Schur algebras and little Schur algebras
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.
scientific article; zbMATH DE number 5614724
scientific article