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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2301.00121 |
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| _version_ | 1866909239818059776 |
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| author | Terwilliger, Paul Žitnik, Arjana |
| author_facet | Terwilliger, Paul Žitnik, Arjana |
| contents | A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner; (ii) each subdiagonal entry is nonzero, and the entry in the top-right corner is nonzero. Let $\mathbb F$ denote a field, and let $V$ denote a nonzero finite-dimensional vector space over $\mathbb F$. We consider an ordered pair of $\mathbb F$-linear maps $A: V \to V$ and $A^*: V \to V$ that satisfy the following two conditions: (i) there exists a basis for $V$ with respect to which the matrix representing $A$ is circular bidiagonal and the matrix representing $A^*$ is diagonal; (ii) there exists a basis for $V$ with respect to which the matrix representing $A^*$ is circular bidiagonal and the matrix representing $A$ is diagonal. We call such a pair a circular bidiagonal pair on $V$. We classify the circular bidiagonal pairs up to affine equivalence. There are two infinite families of solutions, which we describe in detail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00121 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Circular bidiagonal pairs Terwilliger, Paul Žitnik, Arjana Quantum Algebra Combinatorics 17B37 A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner; (ii) each subdiagonal entry is nonzero, and the entry in the top-right corner is nonzero. Let $\mathbb F$ denote a field, and let $V$ denote a nonzero finite-dimensional vector space over $\mathbb F$. We consider an ordered pair of $\mathbb F$-linear maps $A: V \to V$ and $A^*: V \to V$ that satisfy the following two conditions: (i) there exists a basis for $V$ with respect to which the matrix representing $A$ is circular bidiagonal and the matrix representing $A^*$ is diagonal; (ii) there exists a basis for $V$ with respect to which the matrix representing $A^*$ is circular bidiagonal and the matrix representing $A$ is diagonal. We call such a pair a circular bidiagonal pair on $V$. We classify the circular bidiagonal pairs up to affine equivalence. There are two infinite families of solutions, which we describe in detail. |
| title | Circular bidiagonal pairs |
| topic | Quantum Algebra Combinatorics 17B37 |
| url | https://arxiv.org/abs/2301.00121 |