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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Rank-deficient representations in the theta correspondence over finite fields arise from quantum codes
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by Felipe Montealegre-Mora and David Gross PDF
Represent. Theory 25 (2021), 193-223


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
  • Email:
  • David Gross
  • Affiliation: Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany
  • Email:
  • 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).
  • © Copyright 2021 the authors
  • Journal: Represent. Theory 25 (2021), 193-223
  • MSC (2020): Primary 20C33; Secondary 20G40
  • DOI:
  • MathSciNet review: 4235130