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Gaussian maps and plethysm. (English) Zbl 0951.20030
Newstead, Peter E. (ed.), Algebraic geometry. Papers for the EUROPROJ conferences held in Catania, Italy, September 1993, and Barcelona, Spain, September 1994. New York, NY: Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. 200, 91-117 (1998).
Since the irreducible representations of linear groups can be realized as spaces of global holomorphic sections of homogeneous line bundles on flag varieties, in this paper, the author uses Gaussian maps on homogeneous spaces to investigate the properties of plethysm: the stability of its multiplicities. Let \(V\) be an complex vector space, \(\nu=(\nu_1,\ldots,\nu_m)\) be an increasing partition of \(kl\). The multiplicity of the irreducible \(\text{Gl}(V)\)-module \(S^\nu(V)\) of dominant weight \(\nu\) in the plethysm \(S^k(S^l(V))\) is denoted by \(\nu^{k,l}(\mu)\) where \(\mu=(\nu_1,\ldots,\nu_{m-1})\). This multiplicity \(\nu^{k,l}(\mu)\) is known to stabilize when \(l\) becomes large enough.
In this paper, it is shown that stability also occurs for \(k\), that is, \(\nu^{k,l}(\mu)\) is an increasing and stationary function of \(k\) and \(l\), and the \(k\)-stable multiplicity \(\nu^{\infty,l}(\mu)=\lim_{k\to\infty}\nu^{k,l}(\mu)\) equals the multiplicity of \(S^\mu V\) in the sum \[ \bigoplus_{\pi=(2^{\pi_2},\ldots,l^{\pi_l})}S^{\pi_2}(S^2V)\otimes\cdots\otimes S^{\pi_l}(S^lV). \] It is also proved that for a partition \(\mu\) of length less than \(k\) and \(l\geq h(\mu)\), if \(\nu=(\mu,kl-|\mu|)\), then \(S^{2\nu}V\) has positive multiplicity inside \(S^k(S^{2l}V)\). In fact this is Weintraub’s conjecture “after stabilization”.
For the entire collection see [Zbl 0913.00031].

20G05 Representation theory for linear algebraic groups
05E05 Symmetric functions and generalizations
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
14M17 Homogeneous spaces and generalizations