Rank-deficient representations in the theta correspondence over finite fields arise from quantum codes
Felipe Montealegre‐Mora, David R Gross · Representation Theory of the American Mathematical Society · 2021
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 ⊗