Intertwining numbers; the three-rowed case
type: Item
instance of: scholarly article
title: Intertwining numbers; the three-rowed case
zbMATH Open document ID: 0924.20034
author: David A. Buchsbaum
author: Delia Flores de Chela
published in: Journal of Algebra
publication date: 1999-07-16T00:00:00Z
review text: Suppose that \(\lambda\) is a partition, \(\lambda=(\lambda_1,\dots,\lambda_n)\), \(d\) is a positive integer, and \(\mu\) is the partition obtained from \(\lambda\) by taking \(d\) boxes away from some row of \(\lambda\) and attaching them to some higher row (assuming that we still obtain a partition), i.e., \(\mu=(\lambda_1,\dots,\lambda_k+d,\dots,\lambda_{k+j}-d,\dots,\lambda_n)\). A standard question in modular representation theory is the following: What is the \(\mathbb{Z}/(p)\)-dimension of the \(\mathbb{Z}/(p)\)-vector space \(\text{Ext}^i_{A_r}(\overline K_\lambda,\overline K_\mu)\)? (The bars over the modules indicate the reduction of these modules modulo \(p\).) These numbers are called intertwining numbers. The attempts to determine these intertwining numbers are varied. One approach is suggested by the following special case of a theorem found in [\textit{K. Akin} and \textit{D. A. Buchsbaum}, Adv. Math. 72, No. 2, 171-210 (1988; Zbl 0681.20028)]. Let \(M\) and \(N\) be integral polynomial representations of \(\text{Gl}(F)\) of degree \(r\), let \(p\) be a prime, and let \(A_r\) be the integral Schur algebra of degree \(r\). Denote by \(\overline X\) the \(\mathbb{Z}/(p)\)-module (-algebra) \(X\otimes_\mathbb{Z}\mathbb{Z}/(p)\). Then we have the exact sequence \[ 0\to\text{Ext}^i_{A_r}(M,N)\otimes\mathbb{Z}/(p)\to\text{Ext}^i_{\overline A_r}(\overline M,\overline N)\to\text{Tor}^1_\mathbb{Z}(\text{Ext}^{i+1}_{A_r}(M,N),\mathbb{Z}/(p))\to 0. \] From this exact sequence, we see that it suffices to calculate the integral Ext groups, since the modular ones are simply the \(p\)-torsion part of one integral Ext plus the reduction modulo \(p\) of another. A fairly straightforward argument shows that it is enough to consider the case where we take \(d\) boxes from the last row and attach them to the first one. Of course, straight homological algebra tells us that we merely have to find a projective resolution, \(\mathbb{P}\), of \(K_\lambda\) over \(A_r\), and …
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 923700
zbMATH Keywords: partitions
zbMATH Keywords: intertwining numbers
zbMATH Keywords: polynomial representations
zbMATH Keywords: Schur algebras
zbMATH Keywords: projective resolutions
zbMATH Keywords: tensor products
zbMATH Keywords: divided powers
zbMATH Keywords: weights
zbMATH Keywords: cohomology modules
MaRDI profile type: Publication
full work available at URL: jabr.1996.0235
OpenAlex ID: W2023632301
DOI: 10.1006/JABR.1996.0235
description: scientific article; zbMATH DE number 923700