Highest Weight Vectors in Plethysms
- PDF / 477,782 Bytes
- 25 Pages / 439.37 x 666.142 pts Page_size
- 98 Downloads / 193 Views
Communications in
Mathematical Physics
Highest Weight Vectors in Plethysms Kazufumi Kimoto1 , Soo Teck Lee2 1 Department of Mathematical Sciences, University of the Ryukyus, 1 Senbaru, Nishihara,
Okinawa 903-0213, Japan. E-mail: [email protected]
2 Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road,
Singapore 119076, Republic of Singapore. E-mail: [email protected] Received: 17 May 2019 / Accepted: 9 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract: We realize the GLn (C)-modules S k (S m (Cn )) and k (S m (Cn )) as spaces of polynomial functions on n × k matrices. In the case k = 3, we describe explicitly all the GLn (C)-highest weight vectors which occur in S 3 (S m (Cn )) and in 3 (S m (Cn )) respectively. In particular, we obtain alternative formulas for the multiplicities in these modules.
1. Introduction Let V1 , V2 and V3 be finite dimensional complex vector spaces, and let π1 : GL(V1 ) → GL(V2 ) and π2 : GL(V2 ) → GL(V3 ) be polynomial representations of GL(V1 ) and GL(V2 ) respectively. Then the composition π2 ◦ π1 : GL(V1 ) → GL(V3 ) is also a polynomial representation of GL(V1 ). If χ1 and χ2 are the characters of π1 and π2 respectively, then the character of π2 ◦ π1 is called the plethysm of χ1 and χ2 and is denoted by χ2 ◦ χ1 (or χ2 [χ1 ]). One of the main open problems in combinatorics is to express the plethysm sλ ◦ sμ of two Schur polynomials, which are characters of irreducible polynomial representations of the general linear groups with highest weights λ and μ respectively, as a linear combination of Schur polynomials. Let us look at the case of the complete symmetric polynomials, i.e. the plethysms of the form h k ◦ h m . The problem in this case is equivalent to determining the irreducible decomposition of the GLn -module S k (S m (Cn )). Several explicit results are known, for instance, sμ hk ◦ h2 = μ
Kazufumi Kimoto is partially supported by Grant-in-Aid for Scientific Research (C) No. 18K03248, JSPS and by JST CREST Grant Number JPMJCR14D6, Japan. Soo Teck Lee is supported by NUS Grant R-146-000-252-114.
K. Kimoto, S. T. Lee
for an arbitrary positive integer k, where μ runs through all the even partitions of 2k. The formula is actually equivalent to the identity sμ = (1 − xi x j )−1 μ:even
i≤ j
due to Littlewood [Li]. There are also results on h 2 ◦h m , h 3 ◦h m and h 4 ◦h m for an arbitrary positive integer m: the first and second cases are due to Thrall [T] (see also [P]), and the third case is due to Foulkes [Fo]. On the other hand, using representation theory and in particular (GLn , GLm )-duality, Howe [H1] describes the multiplicities in S k (S m (Cn )) for k ≤ 4. An example of recent developments of plethysms is [dBPW], which studies certain stability conditions on the coefficients in the expansion with respect to Schur polynomials by using combinatorics on tableaux whose entries are also tableaux. In this paper, we study plethysms using representation theory and an approach
Data Loading...
