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Rank-deficient representations in the theta correspondence over finite fields arise from quantum codes

Authors: Felipe Montealegre-Mora and David Gross
Journal: Represent. Theory 25 (2021), 193-223
MSC (2020): Primary 20C33; Secondary 20G40
Published electronically: March 25, 2021
MathSciNet review: 4235130
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Abstract: Let $V$ be a symplectic vector space and let $\mu$ be the oscillator representation of $\operatorname {Sp}(V)$. It is natural to ask how the tensor power representation $\mu ^{\otimes t}$ decomposes. If $V$ is a real vector space, then the theta correspondence asserts that there is a one-one correspondence between the irreducible subrepresentations of $\operatorname {Sp}(V)$ 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 $\operatorname {Sp}(V)$ representation. They show that a variant of the Theta correspondence 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 $\operatorname {Sp}(V)$-subrepresentations arise from embeddings of lower-order tensor products of $\mu$ and $\bar \mu$ into $\mu ^{\otimes t}$. 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 $\operatorname {Sp}(V)$-subrepresentations of $\mu ^{\otimes t}$ are labelled by the irreps of orthogonal groups $O(r)$ acting on certain $r$-dimensional spaces for $r\leq t$. The results hold in odd charachteristic and the “stable range” $t\leq \frac 12 \dim V$. 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.

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Additional Information

Felipe Montealegre-Mora
Affiliation: Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany

David Gross
Affiliation: Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany

Received by editor(s): June 24, 2020
Received by editor(s) in revised form: November 23, 2020
Published electronically: March 25, 2021
Additional Notes: This work was by the Excellence Initiative of the German Federal and State Governments (Grant ZUK 81), the ARO under contract W911NF-14-1-0098 (Quantum Characterization, Verification, and Validation), and the DFG (SPP1798 CoSIP, project B01 of CRC 183).
Article copyright: © Copyright 2021 the authors