Representations of \(\text{SL}(2,\mathbb{C})\) and unimodal polynomials.
type: Item
instance of: scholarly article
title: Representations of \(\text{SL}(2,\mathbb{C})\) and unimodal polynomials.
zbMATH Open document ID: 0624.20028
DOI: 10.1016/0021-8693(87)90103-7
author: Gert Almkvist
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: An integral polynomial \(a_ 0+a_ 1t+...+a_ nt^ n\) is called symmetric unimodal if \(a_{n-v}=a_ v\) and \(0<a_ 0\leq a_ 1\leq...\leq a_{[n/2]}\). These polynomials, which arise in various combinatorial problems, are here studied in a new (and natural) way by associating with them certain (finite-dimensional, complex) representations of the group \(G=\text{SL}(2,\mathbb{C})\). Such a representation is called homogeneous if the degrees of its irreducible constituents are either all odd or all even. There is a bijection from the set of all symmetric unimodal polynomials to the set of all homogeneous representations of \(G\), having the property that the image of the product of two polynomials is the tensor product of their images. The irreducible \(G\)-modules may be denoted by \(V_ 1,V_ 2,...\), where \(\dim V_ n=n\). The fact that the exterior power \(\Lambda^ rV_ n\) affords a homogeneous representation yields a new proof that the Gaussian polynomial \(\left[\begin{matrix} n\\ r\end{matrix} \right]\) is symmetric unimodal. More generally, if \(\alpha\) is a partition then the Schur module \(S^{\alpha}V_ n\) affords a homogeneous representation, whence the Gaussian polynomial \(\left[\begin{matrix} n\\ \alpha \end{matrix} \right]\) is, symmetric unimodal. Similar considerations (involving the Adams operations) lead the author to conjecture that \(\prod^{r}_{v=1}(1- t^{vn})/(1-t^ v)\) is symmetric unimodal if \(n\) is even or if \(n\) is odd and \(r\) sufficiently large. This is known to be true for \(n=2\) and the author proves it for \(n=4\), using analytic methods of A. Odlyzko and B. Richmond to estimate the coefficients. Another conjecture put forward is that \((1+t)(1+t^ 3)...(1+t^{2n-1})+t^ 2+t^{n^ 2-2}\) is symmetric unimodal for \(n\geq 27\). The author also uses the (explicitly known) \(p\)-modular representation theory of \(G\) to produce still further symmetric unimodal polynomials. In a note added in proof, the author adverts to his subsequent proof of th…
reviewed by: G. E. Wall
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 4012887
zbMATH Keywords: special linear groups
zbMATH Keywords: irreducible constituents
zbMATH Keywords: symmetric unimodal polynomials
zbMATH Keywords: homogeneous representations
zbMATH Keywords: irreducible modules
zbMATH Keywords: exterior powers
zbMATH Keywords: Gaussian polynomials
zbMATH Keywords: partitions
zbMATH Keywords: Schur modules
zbMATH Keywords: Adams operations
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90103-7
OpenAlex ID: W2090464301
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description: scientific article; zbMATH DE number 4012887