Hook immanantal inequalities for totally nonnegative matrices
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918152288337920 |
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| author | Skandera, Mark |
| author_facet | Skandera, Mark |
| contents | Given a weakly decreasing positive integer sequence $λ= (λ_1,\dotsc,λ_\ell)$ summing to $n$, let $χ^λ$ denote the irreducible character of the symmetric group $S_n$ indexed by $λ$. This representation has dimension $χ^λ(e)$, where $e$ is the identity element of $S_n$. Let $\mathrm{Imm}_{χ^λ}$ denote the corresponding irreducible character immanant, the function on $n \times n$ matrices $A = (a_{i,j})$ defined by $\mathrm{Imm}_{χ^λ}(A) := \sum_{w \in S_n} χ^λ(w) a_{1,w_1} \cdots a_{n,w_n}$. Merris conjectured [Linear Multilinear Algebra 14 (1983) pp. 21--35] and Heyfron proved [Linear Multilinear Algebra 24 (1988) pp. 65--78] that irreducible character immanants indexed by ``hook'' sequences $(k, 1, \dotsc, 1)$ satisfy the inequalities $\mathrm{per}(A)=\frac{\mathrm{Imm}_{χ^n}(A)}{χ^{n}(e)}\geq \frac{\mathrm{Imm}_{χ^{n-1,1}}(A)}{χ^{n-1,1}(e)}\geq \frac{\mathrm{Imm}_{χ^{ n-2,1,1}}(A)}{χ^{n-2,1,1}(e)}\geq \cdots \geq \frac{\mathrm{Imm}_{χ^{1,\dotsc,1}}(A)}{χ^{1,\dotsc,1}(e)}=\det(A)$ whenever $A$ is an $n \times n$ Hermitian positive semidefinite matrix. We prove that the same inequalities hold whenever $A$ is an $n \times n$ totally nonnegative matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hook immanantal inequalities for totally nonnegative matrices Skandera, Mark Combinatorics 05E05, 15B48, 15A15 Given a weakly decreasing positive integer sequence $λ= (λ_1,\dotsc,λ_\ell)$ summing to $n$, let $χ^λ$ denote the irreducible character of the symmetric group $S_n$ indexed by $λ$. This representation has dimension $χ^λ(e)$, where $e$ is the identity element of $S_n$. Let $\mathrm{Imm}_{χ^λ}$ denote the corresponding irreducible character immanant, the function on $n \times n$ matrices $A = (a_{i,j})$ defined by $\mathrm{Imm}_{χ^λ}(A) := \sum_{w \in S_n} χ^λ(w) a_{1,w_1} \cdots a_{n,w_n}$. Merris conjectured [Linear Multilinear Algebra 14 (1983) pp. 21--35] and Heyfron proved [Linear Multilinear Algebra 24 (1988) pp. 65--78] that irreducible character immanants indexed by ``hook'' sequences $(k, 1, \dotsc, 1)$ satisfy the inequalities $\mathrm{per}(A)=\frac{\mathrm{Imm}_{χ^n}(A)}{χ^{n}(e)}\geq \frac{\mathrm{Imm}_{χ^{n-1,1}}(A)}{χ^{n-1,1}(e)}\geq \frac{\mathrm{Imm}_{χ^{ n-2,1,1}}(A)}{χ^{n-2,1,1}(e)}\geq \cdots \geq \frac{\mathrm{Imm}_{χ^{1,\dotsc,1}}(A)}{χ^{1,\dotsc,1}(e)}=\det(A)$ whenever $A$ is an $n \times n$ Hermitian positive semidefinite matrix. We prove that the same inequalities hold whenever $A$ is an $n \times n$ totally nonnegative matrix. |
| title | Hook immanantal inequalities for totally nonnegative matrices |
| topic | Combinatorics 05E05, 15B48, 15A15 |
| url | https://arxiv.org/abs/2510.00327 |