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Register a new userMany zeros of many characters of GL(n,q)
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For $G={rm GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(chi,g)$ in ${rm Irr}(G)times G$ with $chi(g) eq 0$ satisfies $P_{n,q}to 0$ as $ntoinfty$.
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Let $p$ be any prime. Let $P_n$ be a Sylow $p$-subgroup of the symmetric group $S_n$. Let $phi$ and $psi$ be linear characters of $P_n$ and let $N$ be the normaliser of $P_n$ in $S_n$. In this article we show that the inductions of $phi$ and $psi$ to $S_n$ are equal if, and only if, $phi$ and $psi$ are $N$--conjugate. This is an analogue for symmetric groups of a result of Navarro for $p$-solvable groups.
Lascoux stated that the type A Kostka-Foulkes polynomials K_{lambda,mu}(t) expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a t-analogue of the multiplicity of the dominant weight mu in the irreducible representation of highest weight lambda. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K_{lambda,mu}(t). We also define a combinatorial version of the atomic decomposition, as a decomposition of a modified crystal graph. We prove that this stronger version holds in type A (which provides a new, conceptual approach to Lascouxs statement), in types B, C, and D in a stable range for t=1, as well as in some other cases, while we conjecture that it holds more generally. Another conjecture stemming from our work leads to an efficient computation of K_{lambda,mu}(t). We also give a geometric interpretation.
Let $mathsf k$ be a local field. Let $I_ u$ and $I_{ u}$ be smooth principal series representations of $mathrm{GL}_n(mathsf k)$ and $mathrm{GL}_{n-1}(mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for $mathrm{GL}_n(mathsf k)times mathrm{GL}_n(mathsf k)$.
We present sum-sides for principal characters of all standard (i.e., integrable and highest-weight) irreducible modules for the affine Lie algebra $A_2^{(2)}$. We use modifications of five known Bailey pairs; three of these are sufficient to obtain all the necessary principal characters. We then use the technique of Bailey lattice appropriately extended to include out-of-bounds values of one of the parameters, namely, $i$. We demonstrate how the sum-sides break into six families depending on the level of the modules modulo 6, confirming a conjecture of McLaughlin--Sills.
In this note we consider representations of the group GL(n,F), where F is the field of real or complex numbers or, more generally, an arbitrary local field, in the space of equivariant line bundles over Grassmannians over the same field F. We study reducibility and composition series of such representations. Similar results were obtained already in [HL99,Al12,Zel80], but we give a short uniform proof in the general case, using the tools from [AGS15a]. We also indicate some applications to cosine transforms in integral geometry.
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