Staircase Minimality and a Proof of Saxl's Conjecture
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| Format: | Preprint |
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2025
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| _version_ | 1866918436953653248 |
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| author | Lee, Soong Kyum |
| author_facet | Lee, Soong Kyum |
| contents | Saxl's conjecture (2012) asserts that for the staircase partition $ρ_k = (k, k-1, \ldots, 1)$, the tensor square of the corresponding irreducible representation of the symmetric group $S_{T_k}$ contains every irreducible representation as a constituent, where $T_k = k(k+1)/2$ is the $k$th triangular number. We prove this conjecture unconditionally.
Our proof introduces the Staircase Minimality Theorem: among all 2-regular partitions of $T_k$, the staircase $ρ_k$ is the unique dominance-minimal element. Combined with Ikenmeyer's theorem on dominance and Kronecker positivity for staircases, this establishes that every 2-regular partition appears in the tensor square. Modular saturation then follows using only the diagonal entries $d_{μμ} = 1$ of the decomposition matrix, and the Bessenrodt--Bowman--Sutton lifting theorem completes the proof.
We further prove that at triangular numbers, staircases are the only Kronecker-universal self-conjugate partitions, providing a complete characterization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_15035 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Staircase Minimality and a Proof of Saxl's Conjecture Lee, Soong Kyum Representation Theory Combinatorics 20C30 (primary), 05E10, 20C20 (secondary) Saxl's conjecture (2012) asserts that for the staircase partition $ρ_k = (k, k-1, \ldots, 1)$, the tensor square of the corresponding irreducible representation of the symmetric group $S_{T_k}$ contains every irreducible representation as a constituent, where $T_k = k(k+1)/2$ is the $k$th triangular number. We prove this conjecture unconditionally. Our proof introduces the Staircase Minimality Theorem: among all 2-regular partitions of $T_k$, the staircase $ρ_k$ is the unique dominance-minimal element. Combined with Ikenmeyer's theorem on dominance and Kronecker positivity for staircases, this establishes that every 2-regular partition appears in the tensor square. Modular saturation then follows using only the diagonal entries $d_{μμ} = 1$ of the decomposition matrix, and the Bessenrodt--Bowman--Sutton lifting theorem completes the proof. We further prove that at triangular numbers, staircases are the only Kronecker-universal self-conjugate partitions, providing a complete characterization. |
| title | Staircase Minimality and a Proof of Saxl's Conjecture |
| topic | Representation Theory Combinatorics 20C30 (primary), 05E10, 20C20 (secondary) |
| url | https://arxiv.org/abs/2512.15035 |