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

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.