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Register a new userThe Pieri Rule for GLn Over Finite Fields
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The Pieri rule gives an explicit formula for the decomposition of the tensor product of irreducible representation of the complex general linear group GL(n,C) with a symmetric power of the standard representation on C^n. It is an important and long understood special case of the Littlewood-Richardson rule for decomposing general tensor products of representations of GL(n,C). In our recent work [Gurevich-Howe17, Gurevich-Howe19] on the organization of representations of the general linear group over a finite field F_q using small representations, we used a generalization of the Pieri rule to the context of this latter group. In this note, we demonstrate how to derive the Pieri rule for GL(n,Fq). This is done in two steps; the first, reduces the task to the case of the symmetric group S_n, using the natural relation between the representations of S_n and the spherical principal series representations of GL(n,F_q); while in the second step, inspired by a remark of Nolan Wallach, the rule is obtained for S_n invoking the S_ell-GL_(n,C)) Schur duality. Along the way, we advertise an approach to the representation theory of the symmetric group which emphasizes the central role played by the dominance order on Young diagrams. The ideas leading to this approach seem to appear first, without proofs, in [Howe-Moy86].
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We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumars work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule.
The classical Pieri formula gives a combinatorial rule for decomposing the product of a Schur function and a complete homogeneous symmetric polynomial as a linear combination of Schur functions with integer coefficients. We give a Pieri rule for describing the product of an orthosymplectic character and an orthosymplectic character arising from a one-row partition. We establish that the orthosymplectic Pieri rule coincides with Sundarams Pieri rule for symplectic characters and that orthosymplectic characters and symplectic characters obey the same product rule.
There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio trace(pi(g)) / dim(pi), for an irreducible representation pi of G and an element g of G. It turns out [Gurevich-Howe15, Gurevich-Howe17] that for classical groups G over finite fields there are several (compatible) invariants of representations that provide strong information on the character ratios. We call these invariants collectively rank. Rank suggests a new way to organize the representations of classical groups over finite and local fields - a way in which the building blocks are the smallest representations. This is in contrast to Harish-Chandras philosophy of cusp forms that is the main organizational principle since the 60s, and in it the building blocks are the cuspidal representations which are, in some sense, the LARGEST. The philosophy of cusp forms is well adapted to establishing the Plancherel formula for reductive groups over local fields, and led to Lusztigs classification of the irreducible representations of such groups over finite fields. However, analysis of character ratios might benefit from a different approach. In this note we discuss further the notion of tensor rank for GL_n over a finite field F_q and demonstrate how to get information on representations of a given tensor rank using tools coming from the recently studied eta correspondence, as well as the well known philosophy of cusp forms, mentioned just above. A significant discovery so far is that although the dimensions of the irreducible representations of a given tensor rank vary by quite a lot (they can differ by large powers of q), for certain group elements of interest the character ratios of these irreps are nearly equal to each other.
In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field.
Let V be a symplectic vector space and let $mu$ be the oscillator representation of Sp(V). It is natural to ask how the tensor power representation $mu^{otimes t}$ decomposes. If V is a real vector space, then Howe-Kashiwara-Vergne (HKV) duality asserts that there is a one-one correspondence between the irreducible subrepresentations of Sp(V) and the irreps of an orthogonal group 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) representation. They show that a variant of HKV duality 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)-subrepresentations arise from embeddings of lower-order tensor products of $mu$ and $barmu$ into $mu^{otimes t}$. 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) subrepresentations of $mu^{otimes t}$ are labelled by the irreps of orthogonal groups O(r) acting on certain r-dimensional spaces for r <= t. The results hold in odd charachteristic and the stable range t <= 1/2 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.
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