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

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a symplectic vector space and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ<!-- μ --></mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the <italic>oscillator representation</italic> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p left-parenthesis upper V right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Sp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sp}(V)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is natural to ask how the tensor power representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript circled-times t"> <mml:semantics> <mml:msup> <mml:mi>μ<!-- μ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\otimes t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decomposes. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a real vector space, then the theta correspondence asserts that there is a one-one correspondence between the irreducible subrepresentations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p left-parenthesis upper V right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Sp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sp}(V)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the irreps of an orthogonal group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is well-known that this duality fails over finite fields. Addressing this situation, Gurevich and Howe have recently assigned a notion of <italic>rank</italic> to each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p left-parenthesis upper V right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Sp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sp}(V)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p left-parenthesis upper V right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Sp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sp}(V)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subrepresentations arise from embeddings of lower-order tensor products of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ<!-- μ --></mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar \mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript circled-times t"> <mml:semantics> <mml:msup> <mml:mi>μ<!-- μ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\otimes t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The embeddings live on spaces that have been studied in quantum information theory as tensor powers of <italic>self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes</italic>. We then find that the irreducible <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p left-parenthesis upper V right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Sp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sp}(V)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subrepresentations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript circled-times t"> <mml:semantics> <mml:msup> <mml:mi>μ<!-- μ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\otimes t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are labelled by the irreps of orthogonal groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis r right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(r)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acting on certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional spaces for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r less-than-or-equal-to t"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">r\leq t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The results hold in odd charachteristic and the “stable range” <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t less-than-or-equal-to one half dimension upper V"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mi>dim</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t\leq \frac 12 \dim V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our work has implications for the representation theory of the <italic>Clifford group</italic>. 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.