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Rank-deficient representations in the theta correspondence over finite fields arise from quantum codes
Representation Theory ( IF 0.6 ) Pub Date : 2021-03-25 , DOI: 10.1090/ert/563 Felipe Montealegre-Mora , David Gross
Representation Theory ( IF 0.6 ) Pub Date : 2021-03-25 , DOI: 10.1090/ert/563 Felipe Montealegre-Mora , David Gross
Abstract:Let 
be a symplectic vector space and let 
be the oscillator representation of 
. It is natural to ask how the tensor power representation 
decomposes. If 
is a real vector space, then the theta correspondence asserts that there is a one-one correspondence between the irreducible subrepresentations of 
and the irreps of an orthogonal group 
. 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 
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 
-subrepresentations arise from embeddings of lower-order tensor products of 
and 
into 
. 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 
-subrepresentations of 
are labelled by the irreps of orthogonal groups 
acting on certain 
-dimensional spaces for 
. The results hold in odd charachteristic and the ``stable range'' 
. 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.
中文翻译：
有限域中theta对应关系中的秩不足表示法是由量子码产生的
摘要：设是辛向量空间，让是振荡器表示的。很自然地问张量幂表示如何分解。如果是实向量空间，则theta对应关系断言在的不可约子表示与正交组的irrep之间存在一一对应关系。众所周知，这种对偶性在有限域上失效。针对这种情况，Gurevich和Howe最近为每个人分配了等级概念 表示。他们表明，如果将Theta对应关系的一种变体继续限制在有限域上，即使人们将注意力集中在最大秩的子表示上。排名不足的组件的性质是开放的。在这里，我们表明，所有秩亏-subrepresentations从较低阶张量产品的嵌入出现和进入。嵌入生活在空间中，该空间在量子信息论中已被研究为自正交Calderbank-Shor-Steane（CSS）量子码的张量。然后，我们发现的不可约子表示由作用于特定维空间的正交基团的irrep标记为 。结果具有奇特特征和``稳定范围'' 。我们的工作对克利福德集团的表征理论具有启示意义。可以认为这是根据自对偶编码对Clifford组不变量的已知特征的概括。
更新日期：2021-03-25
be a symplectic vector space and let be the oscillator representation of . It is natural to ask how the tensor power representation decomposes. If is a real vector space, then the theta correspondence asserts that there is a one-one correspondence between the irreducible subrepresentations of and the irreps of an orthogonal group . 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 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 -subrepresentations arise from embeddings of lower-order tensor products of and into . 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 -subrepresentations of are labelled by the irreps of orthogonal groups acting on certain -dimensional spaces for . The results hold in odd charachteristic and the ``stable range'' . 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. 中文翻译：
有限域中theta对应关系中的秩不足表示法是由量子码产生的
摘要：设
是辛向量空间，让是振荡器表示的。很自然地问张量幂表示如何分解。如果是实向量空间，则theta对应关系断言在的不可约子表示与正交组的irrep之间存在一一对应关系。众所周知，这种对偶性在有限域上失效。针对这种情况，Gurevich和Howe最近为每个人分配了等级概念 表示。他们表明，如果将Theta对应关系的一种变体继续限制在有限域上，即使人们将注意力集中在最大秩的子表示上。排名不足的组件的性质是开放的。在这里，我们表明，所有秩亏-subrepresentations从较低阶张量产品的嵌入出现和进入。嵌入生活在空间中，该空间在量子信息论中已被研究为自正交Calderbank-Shor-Steane（CSS）量子码的张量。然后，我们发现的不可约子表示由作用于特定维空间的正交基团的irrep标记为 。结果具有奇特特征和``稳定范围'' 。我们的工作对克利福德集团的表征理论具有启示意义。可以认为这是根据自对偶编码对Clifford组不变量的已知特征的概括。 
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