The behavior of Riesz means of the coefficients of a symmetric square L-function

Let f(z) be a holomorphic Hecke eigencuspform of even weight k with respect to SL(2, Z) and let L(s, sym ² f) = ∑ _n=1 ^∞ c_nn^−s, Re s > 1, be the symmetric square L-function associated with f. Represent the Riesz mean (ρ ≥ 0)

Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form

Let p ≥ 3 be a prime and F a totally real number field. Let f be a Hilbert cuspidal eigenform of parallel weight 2, trivial Nebentypus and ordinary at p. It is possible to construct a p-adic L function which interpolates the complex L-function associated with the symmetric square representation of f. This p-adic L-function vanishes at s = 1 even if the complex L-function does not. Assuming p inert and f Steinberg at p, we give a formula for the p-adic derivative at s = 1 of this p-adic L-function, generalizing unpublished work of Greenberg and Tilouine. Under some hypotheses on the conductor of f we prove a particular case of a conjecture of Greenberg on trivial zeros.     

Tensor square of the basic spin representations of Schur covering groups for the symmetric groups

Journal of Algebraic Combinatorics - We consider the tensor square of the basic spin representations of Schur covering groups $$widetilde{S_n}$$ and $$widetilde{S_n^{'}}$$ for the symmetric...     

Combinatorics,Bethe Ansatz,and representations of the symmetric group

Techniques developed in the realms of the quantum method of the inverse problem are used to analyze combinatorial problems (Young diagrams and rigged configurations).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 50–64, 1986.     

Pfaffians and representations of the symmetric group

Pfaffians of matrices with entries z[i, j]/（xi ＋ xj）, or determinants of matrices with entries z[i, j]/（xi - xj）, where the antisymmetrical indeterminates z[i, j] satisfy the Pliicker relations, can be identified with a trace in an irreducible representation of a product of two symmetric groups. Using Young＇s orthogonal bases, one can write explicit expressions of such Pfaffians and determinants, and recover in particular the evaluation of Pfaffians which appeared in the recent literature.     

Distance-transitive representations of the symmetric groups

Let Γ be a graph and G ≤ Aut(Γ). The group G is said to act distance-transitively on Γ if, for any vertices x, y, u, v such that ∂(x, y) = ∂(u, v), there is an element g ϵ G mapping x into u and y into v. If G acts distance-transitively on Γ then the permutation group induced by the action of G on the vertex set of Γ is called the distance-transitive representation of G. In the paper all distance-transitive representations of the symmetric groups S_n are classified. Moreover, all pairs (G, Γ) such that G acts distance-transitively on Γ and G = S_n for some n are described. The classification problem for these pairs was posed by N. Biggs (Ann. N.Y. Acad. Sci. 319 (1979), 71–81). The problem is closely related to the general question about distance-transitive graphs with given automorphism group.     

On the symmetric square Unstable twisted characters

We provide a purely local computation of the (elliptic) twisted (by “transpose-inverse”) character of the representationπ=I(1) of PGL(3) over ap-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads — see [FK] — to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals. The new case in [FK] is the unstable one. A direct local proof of the fundamental lemma is given in [V].     

QUASI-PERMUTATION REPRESENTATIONS OF ALTERNATING AND SYMMETRIC GROUPS

The quantities c(G), q(G) and p(G) for finite groups were defined by H. Behravesh. In this article, these quantities for the alternating group An and the symmetric group Sn are calculated. It is shown that c(G) = q(G)=p(G) = n, when G = An or Sn.     

Modular forms and representations of symmetric groups

We give an interpretation of the coefficients of some modular forms in terms of modular representations of symmetric groups. Using this we can obtain asymptotic formulas for the number of blocks of the symmetric group S_n over a field of characteristic p for n . For p < 7 we give simple explicit formulas for the number of blocks of defect zero. The study of the modular forms leads to interesting identities involving the Dedekind -function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 74–85, 1982.     

On the symmetric square: unstable local transfer

Summary We prove the fundamental lemma for spherical functions with respect to the natural (induction) lifting fromPGL(2) toPGL(3) which appears as the unstable counterpart of the stable symmetric-square lifting fromSL(2) toPGL(3) (see [IV] for an introduction to this project, and [VI] for the final results). Thus spherical functions onPGL(2) andPGL(3) which correspond to each other by satisfying an elementary representation theoretic relation are shown to have matching orbital integrals. The proof of this local statement is based on an application of the global trace formula.Partially supported by NSF grants     

On the central value of symmetric square L-functions

Let S _k (N, χ) be the space of cusp forms of weight k, level N and character χ. For let L(s, sym² f) be the symmetric square L-function and be the Rankin–Selberg square attached to f. For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym² f) and over a basis of S _k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym² f) does not vanish. The author was supported by NSERC grant 311664-05.     

Relations between the Clausen and Kazhdan-Lusztig representations of the symmetric group

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x_1,1,…,x_n,n] to give a new construction of the Kazhdan-Lusztig representations of S_n. This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x_1,1,…,x_n,n]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young’s natural representations.