Integral Ext 2 Between Hook Weyl Modules
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Integral Ext2 Between Hook Weyl Modules Dimitra-Dionysia Stergiopoulou1 Received: 19 June 2020 / Accepted: 9 October 2020 / © Springer Nature B.V. 2020
Abstract This paper concerns representations of the integral general linear group. The extension groups Ext2 between any pair of hook Weyl modules are determined via a detailed study of cyclic generators and relations associated to certain extensions. As a corollary, the modular extension groups Ext1 between such modules are determined. Keywords Extensions · Weyl modules · General linear group · Hook Mathematics Subject Classification (2010) Primary 20G05
1 Introduction In the study of polynomial representations of the integral general linear group GLn , the Weyl modules (λ), indexed by partitions λ, play a central role. These modules have an explicit construction, enjoy a standard basis theorem and their characters are the classical Schur functions [10]. Moreover, the heads of the modules K ⊗ (λ) form a complete set of inequivalent polynomial simple modules for GLn (K), where K is an infinite field. However, the structure of the modules (λ) is not well understood. One of the important problems in the area is to determine the extension groups Exti ((λ), (μ)). Relatively few explicit results are known. Modular extension groups for GLn (K) were studied in [1], when λ consists of a single column and μ of a single row. For SL2 (K), all modular extension groups between Weyl modules were described in [16], generalizing [9] and [7]. The modular extension groups Ext1 (K ⊗ (λ), K ⊗ (μ)) for GLn (K) were determined in [8], when μ consists of a single row. Concerning the integral extension groups, Ext1 ((λ), (μ)) for GL3 was computed in [6], when λ and μ differ by a multiple of a positive root. The groups Ext1 ((λ), (μ)) for GLn were determined in [13], see also [4] for a different proof, when λ and μ differ by a single positive root. The groups Exti ((h), (h(k))) were computed in [15] for i = 1, k and
Presented by: Michela Varagnolo Dimitra-Dionysia Stergiopoulou
[email protected] 1
Department of Mathematics, University of Athens, Athens, Greece
D.-D. Stergiopoulou
any pair of hooks h = (a, 1b ) and h(k) = (a+k, 1b−k ). The main method there was to determine cyclic generators of certain extension groups of the form Exti ((h), Da+k ⊗ b−k ), where D∗ and ∗ denote the divided power algebra and the exterior algebra, respectively, of the natural GLn -module. Except for some extreme cases, these generators usually have an involved form. The next step in our method was to calculate the images of these under canonical maps. The idea of using the ‘skew’ extensions Exti ((h), Da+k ⊗b−k ), in place of the usual extensions Exti ((h), (h(k))), comes from the observation that the former seem to have more manageable presentation matrices. The reason for this is that there is no straightening law involved. The purpose of the present paper is to study Ext2 between any two hook Weyl modules using the above strategy. The main result is the following. Theorem 1 C
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