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

Abstract

Let V V be a symplectic vector space and let μ \mu be the oscillator representation of Sp ⁡ ( V ) \operatorname {Sp}(V) . It is natural to ask how the tensor power representation μ ⊗ t \mu ^{\otimes t} decomposes. If V V is a real vector space, then the theta correspondence asserts that there is a one-one correspondence between the irreducible subrepresentations of Sp ⁡ ( V ) \operatorname {Sp}(V) and the irreps of an orthogonal group O ( t ) 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 ) \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 Sp ⁡ ( V ) \operatorname {Sp}(V) -subrepresentations arise from embeddings of lower-order tensor products of μ \mu and μ ¯ \bar \mu into μ ⊗ t \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 Sp ⁡ ( V ) \operatorname {Sp}(V) -subrepresentations of μ ⊗ t \mu ^{\otimes t} are labelled by the irreps of orthogonal groups O ( r ) O(r) acting on certain r r -dimensional spaces for r ≤ t r\leq t . The results hold in odd charachteristic and the “stable range” t ≤ 1 2 dim ⁡ V 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.