Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T03:25:22.692Z Has data issue: false hasContentIssue false

On Tensor Products of Polynomial Representations

Published online by Cambridge University Press:  20 November 2018

Kevin Purbhoo
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2. e-mail: [email protected] e-mail: [email protected]
Stephanie van Willigenburg
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2. e-mail: [email protected] e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $\text{GL}\left( n,\,\mathbb{C} \right)$ is isomorphic to another. As a consequence we discover families of Littlewood–Richardson coefficients that are non-zero, and a condition on Schur non-negativity.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Fomin, S., Fulton, W., Li, C., and Poon, Y., Eigenvalues, singular values, and Littlewood-Richardson coefficients. Amer. J. Math. 127(2005), no. 1, 101127.Google Scholar
[2] Macdonald, I., Symmetric Functions and Hall Polynomials. Second edition. Oxford University Press, New York, 1995.Google Scholar
[3] Lascoux, A., Leclerc, B., and Thibon, J.-Y., Ribbon tableaux, Hall-Littlewood functions, quantum affine algebras and unipotent varieties. J. Math. Phys. 38(1997), no. 2, 10411068.Google Scholar
[4] Lam, T., Postnikov, A., and Pylyavskyy, P., Schur positivity and Schur log-concavity. Amer. J. Math. 129(2007), no. 6, 16111622.Google Scholar
[5] Okounkov, A., Log-concavity of multiplicities with applications to characters of U(). Adv. Math. 127(1997), no. 2, 258282.Google Scholar
[6] Rajan, C., Unique decomposition of tensor products of irreducible representations of simple algebraic groups. Ann. of Math. 160(2004), no. 2, 683704.Google Scholar
[7] Rhoades, B. and Skandera, M., Kazhdan-Lusztig immanants and products of matrix minors. J. Algebra 304(2006), no. 2, 793811.Google Scholar
[8] Stanley, R., Enumerative Combinatorics. Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge, 1999.Google Scholar