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

Let V V be a symplectic vector space and let alttext="mu"> μ encoding="application/x-tex">\mu the oscillator representation of S p left-parenthesis upper V right-parenthesis"> Sp ⁡ ( stretchy="false">) encoding="application/x-tex">\operatorname {Sp}(V) . It is natural to ask how tensor power representation alttext="mu Superscript circled-times t"> ⊗<!-- ⊗ <mml:mi>t encoding="application/x-tex">\mu ^{\otimes t} decomposes. If real space, then theta correspondence asserts that there one-one between irreducible subrepresentations irreps an orthogonal group O t O encoding="application/x-tex">O(t) well-known this duality fails over finite fields. Addressing situation, Gurevich Howe have recently assigned notion rank each representation. They show variant Theta continues hold fields, if one restricts attention maximal rank. The nature rank-deficient components was left open. Here, we all -subrepresentations arise from embeddings lower-order products overbar"> stretchy="false">¯<!-- ¯ </mml:mover> encoding="application/x-tex">\bar \mu into live on spaces been studied in quantum information theory as powers self-orthogonal Calderbank-Shor-Steane (CSS) codes. We find are labelled by groups r r encoding="application/x-tex">O(r) acting certain alttext="r"> encoding="application/x-tex">r -dimensional for alttext="r less-than-or-equal-to ≤<!-- ≤ encoding="application/x-tex">r\leq t results odd charachteristic “stable range” alttext="t half dimension 1 2 dim encoding="application/x-tex">t\leq \frac 12 \dim V Our work has implications Clifford group. can thought generalization known characterization invariants Clifford terms self-dual codes.