Geometric and Combinatorial Properties of the Alternating Sign Matrix Polytope
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910875340767232 |
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| author | Dinkelman, Elizabeth A. Morris Jr, Walter D. |
| author_facet | Dinkelman, Elizabeth A. Morris Jr, Walter D. |
| contents | The polytope $ASM_n$, the convex hull of the $n\times n$ alternating sign matrices, was introduced by Striker and by Behrend and Knight. A face of $ASM_n$ corresponds to an elementary flow grid defined by Striker, and each elementary flow grid determines a doubly directed graph defined by Brualdi and Dahl. We show that a face of $ASM_n$ is symmetric if and only if its doubly directed graph has all vertices of even degree. We show that every face of $ASM_n$ is a 2-level polytope. We show that a $d$-dimensional face of $ASM_n$ has at most $2^d$ vertices and $4(d-1)$ facets, for $d\ge 2$. We show that a $d$-dimensional face of $ASM_n$ satisfies $vf\le d2^{d+1}$, where $v$ and $f$ are the numbers of vertices and edges of the face. If the doubly directed graph of a $d$-dimensional face is 2-connected, then $v\le 2^{d-1}+2$. We describe the facets of a face and a basis for the subspace parallel to a face in terms of the elementary flow grid of the face. We prove that no face of $ASM_n$ has the combinatorial type of the Birkhoff polytope $B_3$. We list the combinatorial types of faces of $ASM_n$ that have dimension 4 or less. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric and Combinatorial Properties of the Alternating Sign Matrix Polytope Dinkelman, Elizabeth A. Morris Jr, Walter D. Combinatorics 52B05 The polytope $ASM_n$, the convex hull of the $n\times n$ alternating sign matrices, was introduced by Striker and by Behrend and Knight. A face of $ASM_n$ corresponds to an elementary flow grid defined by Striker, and each elementary flow grid determines a doubly directed graph defined by Brualdi and Dahl. We show that a face of $ASM_n$ is symmetric if and only if its doubly directed graph has all vertices of even degree. We show that every face of $ASM_n$ is a 2-level polytope. We show that a $d$-dimensional face of $ASM_n$ has at most $2^d$ vertices and $4(d-1)$ facets, for $d\ge 2$. We show that a $d$-dimensional face of $ASM_n$ satisfies $vf\le d2^{d+1}$, where $v$ and $f$ are the numbers of vertices and edges of the face. If the doubly directed graph of a $d$-dimensional face is 2-connected, then $v\le 2^{d-1}+2$. We describe the facets of a face and a basis for the subspace parallel to a face in terms of the elementary flow grid of the face. We prove that no face of $ASM_n$ has the combinatorial type of the Birkhoff polytope $B_3$. We list the combinatorial types of faces of $ASM_n$ that have dimension 4 or less. |
| title | Geometric and Combinatorial Properties of the Alternating Sign Matrix Polytope |
| topic | Combinatorics 52B05 |
| url | https://arxiv.org/abs/2503.10862 |