Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2019
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| _version_ | 1866917977856671744 |
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| author | Montealegre-Mora, Felipe Gross, David |
| author_facet | Montealegre-Mora, Felipe Gross, David |
| contents | Let V be a symplectic vector space and let $μ$ be the oscillator representation of Sp(V). It is natural to ask how the tensor power representation $μ^{\otimes t}$ decomposes. If V is a real vector space, then Howe-Kashiwara-Vergne (HKV) duality asserts that there is a one-one correspondence between the irreducible subrepresentations of Sp(V) and the irreps of an orthogonal group O(t). It is well-known that this duality fails over finite fields. Addressing this situation, Gurevich and Howe have recently assigned a notion of rank to each Sp(V) representation. They show that a variant of HKV duality continues to hold over finite fields, if one restricts attention to subrepresentations of maximal rank. The nature of the rank-deficient components was left open. Here, we show that all rank-deficient Sp(V)-subrepresentations arise from embeddings of lower-order tensor products of $μ$ and $\barμ$ into $μ^{\otimes t}$. The embeddings live on spaces that have been studied in quantum information theory as tensor powers of self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes. We then find that the irreducible Sp(V) subrepresentations of $μ^{\otimes t}$ are labelled by the irreps of orthogonal groups O(r) acting on certain r-dimensional spaces for r <= t. The results hold in odd charachteristic and the "stable range" t <= 1/2 dim V. Our work has implications for the representation theory of the Clifford group. It can be thought of as a generalization of the known characterization of the invariants of the Clifford group in terms of self-dual codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_07230 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes Montealegre-Mora, Felipe Gross, David Representation Theory Mathematical Physics Quantum Physics Let V be a symplectic vector space and let $μ$ be the oscillator representation of Sp(V). It is natural to ask how the tensor power representation $μ^{\otimes t}$ decomposes. If V is a real vector space, then Howe-Kashiwara-Vergne (HKV) duality asserts that there is a one-one correspondence between the irreducible subrepresentations of Sp(V) and the irreps of an orthogonal group O(t). It is well-known that this duality fails over finite fields. Addressing this situation, Gurevich and Howe have recently assigned a notion of rank to each Sp(V) representation. They show that a variant of HKV duality continues to hold over finite fields, if one restricts attention to subrepresentations of maximal rank. The nature of the rank-deficient components was left open. Here, we show that all rank-deficient Sp(V)-subrepresentations arise from embeddings of lower-order tensor products of $μ$ and $\barμ$ into $μ^{\otimes t}$. The embeddings live on spaces that have been studied in quantum information theory as tensor powers of self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes. We then find that the irreducible Sp(V) subrepresentations of $μ^{\otimes t}$ are labelled by the irreps of orthogonal groups O(r) acting on certain r-dimensional spaces for r <= t. The results hold in odd charachteristic and the "stable range" t <= 1/2 dim V. Our work has implications for the representation theory of the Clifford group. It can be thought of as a generalization of the known characterization of the invariants of the Clifford group in terms of self-dual codes. |
| title | Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes |
| topic | Representation Theory Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/1906.07230 |