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Asymptotic and catalytic containment of representations of SU(n)
Fritz, Tobias1
1 University of Innsbruck Department of Mathematics Technikerstr. 13 6020 Innsbruck Austria
Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 327-336

Given two finite-dimensional representations ρ and σ of SU(n), when is there n∈ℕ such that ρ ⊗n is isomorphic to a subrepresentation of σ ⊗n ? When is there a third representation η such that ρ⊗η is a subrepresentation of σ⊗η? We call these the questions of asymptotic and catalytic containment, respectively.

We answer both questions in terms of an explicit family of inequalities. These inequalities are generically necessary and sufficient in the following sense. If two representations satisfy all inequalities strictly, then asymptotic and catalytic containment follow (the former in generic cases). Conversely, if asymptotic or catalytic containment holds, then the inequalities must hold non-strictly. These results are an instance of a recent Vergleichsstellensatz applied to the representation semiring.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.338
Classification: 20G05, 19A22, 06F25
Keywords: Schur positivity, Schur polynomial, asymptotic representation theory, $\mathrm{SL}(n,\mathbb{C})$, $\mathrm{SU}(n)$, preordered semiring, Vergleichsstellensatz

Fritz, Tobias 1

1 University of Innsbruck Department of Mathematics Technikerstr. 13 6020 Innsbruck Austria
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Fritz, Tobias. Asymptotic and catalytic containment of representations of $\mathrm{SU}(n)$. Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 327-336. doi: 10.5802/alco.338

[1] Aissen, Michael; Edrei, Albert; Schoenberg, I. J.; Whitney, Anne On the generating functions of totally positive sequences, Proc. Nat. Acad. Sci. U.S.A., Volume 37 (1951), pp. 303-307 | DOI | MR | Zbl

[2] Aissen, Michael; Schoenberg, I. J.; Whitney, A. M. On the generating functions of totally positive sequences. I, J. Analyse Math., Volume 2 (1952), pp. 93-103 | DOI | MR

[3] Brylawski, Thomas The lattice of integer partitions, Discrete Math., Volume 6 (1973), pp. 201-219 | DOI | MR

[4] Fritz, Tobias Characterizing the asymptotic and catalytic stochastic orders on topological abelian groups, 2020 (to appear in Bernoulli) | arXiv

[5] Fritz, Tobias Abstract Vergleichsstellensätze for preordered semifields and semirings I, SIAM J. Appl. Algebra Geom., Volume 7 (2023) no. 2, pp. 505-547 | DOI | MR | Zbl

[6] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR

[7] Székelyhidi, László; Vajday, László Functional equations on the SU(2)-hypergroup, Math. Pannon., Volume 23 (2012) no. 2, pp. 187-193 | MR | Zbl

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