A Short Proof for the Polynomiality of the Stretched Littlewood-Richardson Coefficients
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909247659311104 |
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| author | Thawinrak, Warut |
| author_facet | Thawinrak, Warut |
| contents | The stretched Littlewood-Richardson coefficient $c^{tν}_{tλ,tμ}$ was conjectured by King, Tollu, and Toumazet to be a polynomial function in $t.$ It was shown to be true by Derksen and Weyman using semi-invariants of quivers. Later, Rassart used Steinberg's formula, the hive conditions, and the Kostant partition function to show a stronger result that $c^ν_{λ,μ}$ is indeed a polynomial in variables $ν, λ, μ$ provided they lie in certain polyhedral cones. Motivated by Rassart's approach, we give a short alternative proof of the polynomiality of $c^{tν}_{tλ,tμ}$ using Steinberg's formula and a simple argument about the chamber complex of the Kostant partition function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_06810 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Short Proof for the Polynomiality of the Stretched Littlewood-Richardson Coefficients Thawinrak, Warut Combinatorics The stretched Littlewood-Richardson coefficient $c^{tν}_{tλ,tμ}$ was conjectured by King, Tollu, and Toumazet to be a polynomial function in $t.$ It was shown to be true by Derksen and Weyman using semi-invariants of quivers. Later, Rassart used Steinberg's formula, the hive conditions, and the Kostant partition function to show a stronger result that $c^ν_{λ,μ}$ is indeed a polynomial in variables $ν, λ, μ$ provided they lie in certain polyhedral cones. Motivated by Rassart's approach, we give a short alternative proof of the polynomiality of $c^{tν}_{tλ,tμ}$ using Steinberg's formula and a simple argument about the chamber complex of the Kostant partition function. |
| title | A Short Proof for the Polynomiality of the Stretched Littlewood-Richardson Coefficients |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2211.06810 |