Irreducible Representations Of GL Research Articles

Jacobi–Trudi type formula for a class of irreducible representations of 𝔤𝔩(m|n)

We prove a determinantal type formula to compute the characters of a class of finite-dimensional irreducible representations of the general Lie super-algebra [Formula: see text] in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula, originally conjectured by van der Jeugt and Moens, can be regarded as a generalization of the well-known Jacobi–Trudi formula.

Pieri Type Rules and GL(2|2) Tensor Products

We derive a closed formula for the tensor product of a family of mixed tensors using Deligne’s interpolating category underline {Rep}(GL_{0}). We use this formula to compute the tensor product of a family of irreducible GL(n|n)-representations. This includes the tensor product of any two maximal atypical irreducible representations of GL(2|2).

Local deformation rings for GL2 and a Breuil–Mézard conjecture when l≠p

We compute the deformation rings of two dimensional mod l representations of Gal(Fbar/F) with fixed inertial type, for l an odd prime, p a prime distinct from p and F/Q_p a finite extension. We show that in this setting (when p is also odd) an analogue of the Breuil-M\'{e}zard conjecture holds, relating the special fibres of these deformation rings to the mod l reduction of certain irreducible representations of GL_2(O_F).

On the local Bump–Friedberg L-function

Abstract Let F be a p-adic field. If π be an irreducible representation of GL ( n , F ) ${{\rm GL}(n,F)}$ , Bump and Friedberg associated to π an Euler factor L ( π , 𝐵𝐹 , s 1 , s 2 ) ${L(\pi ,\mathit {BF},s_1,s_2)}$ in [Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday. Part II, Weizmann, Jerusalem (1990), 47–65], that should be equal to L ( φ ( π ) , s 1 ) L ( φ ( π ) , Λ 2 , s 2 ) ${L(\phi (\pi ),s_1)L(\phi (\pi ),\Lambda ^2,s_2)}$ , where φ ( π ) ${\phi (\pi )}$ is the Langlands' parameter of π. The main result of this paper is to show that this equality is true when ( s 1 , s 2 ) = ( s + 1 / 2 , 2 s ) $(s_1,s_2)=(s+1/2,2s)$ , for s in ℂ. To prove this, we classify in terms of distinguished discrete series, generic representations of GL ( n , F ) ${{\rm GL}(n,\hspace*{-0.56905pt}F)}$ which are χ α ${\chi _\alpha }$ -distinguished by the Levi subgroup GL ( [ ( n + 1 ) / 2 ] , F ) × GL ( [ n / 2 ] , F ) ${\rm GL}([(n+1)/2], \hspace*{-0.56905pt}F) \times {\rm GL}([n/2], \hspace*{-0.56905pt}F)$ , for χ α ( g 1 , g 2 ) = α ( det ( g 1 ) / det ( g 2 ) ) ${\chi _\alpha (g_1,g_2)=\alpha (\det (g_1)/{\det }(g_2))}$ , where α is a character of F* of real part between - 1 / 2 ${-1/2}$ and 1/2. We then adapt the technique of [`Derivatives and L-functions for GL ( n ) ${\rm GL}(n)$ ', preprint 2011] to reduce the proof of the equality to the case of discrete series. The equality for discrete series is a consequence of the relation between linear periods and Shalika periods for discrete series, and the main result of [Math. Res. Lett. 19 (2012), no. 4, 785–804].

Endoscopy and the transfer from GSp(4) to GL(4)

Abstract We prove Jiang's conjecture (1996) to show the existence of an L-function of a cuspidal representation of GSp(4,𝔸) × GSp(4,𝔸) which has a pole of order 2 at s = 1, even for globally generic representations. Moreover, we show that if a cuspidal representation π of GSp(4,𝔸) comes from GSO(4,𝔸), then π is the Weil transfer of Π1 ⊗ Π2 realized as a representation of GSO(4,𝔸) where the Π i are irreducible automorphic cuspidal representations of GL(2,𝔸). This agrees with the Langlands functoriality principle as GSO(4,𝔸) is an endoscopic group for GSp(4,𝔸).

Trilinear Forms and Triple Product Epsilon Factors

We give a short and simple proof of a theorem of Dipendra Prasad on the existence or nonexistence of invariant trilinear forms on a triple of irreducible representations of GL 2 (F) or D × , where $F$ is a nonarchimedean local field of zero or odd characteristic and $D$ is the unique quaternion division F-algebra. The answer is controlled by the central value of the triple product epsilon factor. Our proof works uniformly for all representations and without restriction on residual characteristic. It also gives an analogous theorem for any separable cubic F-algebra.

COHOMOLOGICAL BRST ASPECTS OF THE MASSLESS TENSOR FIELD WITH THE MIXED SYMMETRY (k,k)

The main BRST cohomological properties of a free, massless tensor field that transforms in an irreducible representation of GL(D,ℝ), corresponding to a rectangular, two-column Young diagram with k>2 rows are studied in detail. In particular, it is shown that any nontrivial co-cycle from the local BRST cohomology group H(s|d) can be taken to stop either at antighost number (k+1) or k, its last component belonging to the cohomology of the exterior longitudinal derivative H(γ) and containing nontrivial elements from the (invariant) characteristic cohomology H inv (δ|d).

On the restriction of cuspidal representations to unipotent elements

Let G be a connected split reductive group defined over a finite field [ ]q, and G([ ]q) the group of [ ]q-rational points of G. For each maximal torus T of G defined over [ ]q and a complex linear character θ of T([ ]q), let RGT(θ) be the generalized representation of G([ ]q) defined in [DL]. It can be seen that the conjugacy classes in the Weyl group W of G are in one-to-one correspondence with the conjugacy classes of maximal tori defined over [ ]q in G ([C1, 3·3·3]). Let c be the Coxeter conjugacy class of W, and let Tc be the corresponding maximal torus. Then by [DL] we know that πθ = (−1)nRGTc(θ) (where n is the semisimple rank of G and θ is a character in ‘general position’) is an irreducible cuspidal representation of G([ ]q). The results of this paper generalize the pattern about the dimensions of cuspidal representations of GL(n, [ ]q) as an alternating sum of the dimensions of certain irreducible representations of GL(n, [ ]q) appearing in the space of functions on the flag variety of GL(n, [ ]q) as shown in the table below.

Littlewood-Richardson coefficients via Yang-Baxter equation

The purpose of this paper is to present an interpretation for the decomposition of the tensor product of two or more irreducible representations of GL(N) in terms of a system of quantum particles. Our approach is based on a certain scattering matrix that satisfies a Yang-Baxter type equation. The corresponding piecewise-linear transformations of parameters give a solution to the tetrahedron equation. These transformation maps are naturally related to the dual canonical bases for modules over the quantum enveloping algebra Uq(sln)⁠. A byproduct of our construction is an explicit description for the cone of Kashiwara's parametrizations of dual canonical bases. This solves a problem posed by Berenstein and Zelevinsky. We present a graphical interpretation of the scattering matrices in terms of web functions, which are related to honeycombs of Knutson and Tao.

New non recursive formulas for irreducible representations of GL(C m +1 ) and systems of equations for the symmetric powers Sym n P m

New non recursive formulas for irreducible representations of GL(C ^{<i>m</i> +1} ) and systems of equations for the symmetric powers Sym ^<i>n</i> P <i>m</i>

Stability of the Poincaré Bundle

AbstractLet C be a nonsingular projective curve of genus g ≥2 defined over the complex numbers, and let Mξ denote the moduli space of stable bundles of rank n and determinant ξ on C, where ξ is a line bundle of degree don C and n and d are coprime. It is shown that a universal bundle Uξ on C × Mξ is stable with respect to any polarisation on C × Mξ. Similar results are obtained for the case where the determinant is not fixed and for the bundles associated to the universal bundles by irreducible representations of GL(n, C). It is shown further that the connected component of the moduli space of bundles with the same Hilbert polynomial as Uξ on C × Mξ containing Uξ is isomorphic to the Jacobian of C.

Applications de Gauss et pléthysme

The irreducible representations of Gl (n,ℂ) can be described by Schur functors, the composition of which defines plethysm. Its understanding is an important problem of invariant theory, as well as in relation with the representations of symmetric groups.

Correspondence on characters of irreducible unitary representations ofGL(n,ℂ)

Correspondence on characters of irreducible unitary representations ofGL(n,ℂ)

On resolving the multiplicity of the branching rule GL(2k+1,C) down arrow SO(2k+1,C)

We consider the multiplicity problem of the branching rule GL(2k+1,C) down arrow SO(2k+1,C). Finite-dimensional irreducible representations of GL(2k+1,C) are realized as right translations on subspaces of the holomorphic Hilbert (Bargmann) spaces of q*(2k+1) complex variables. Maps are exhibited which carry an irreducible representation of SO(2k+1,C) into these subspaces. An algebra of commuting operators is constructed. Eigenvalues and eigenvectors of certain of these operators can then be used to resolve the multiplicity in the branching rule.

Notes on representations of non-Archimedean SL(n)

Let F be a non-archimedean local field. In this paper the relation between irreducible representations of GL(n, F) and SL(«, F) is studied. Using the results on GL(n, F) a parametrization of (var- ious classes of) irreducible representations of SL(n, F) by parame- ters expressed in terms of cuspidal representations of GL(n, F) is obtained. Introduction. Before we give a more detailed description of the con- tent of this paper, a few historical remarks on SL(n, F) are needed. Gelfand and Naimark gave in (8) proof of the irreducibility of unitary principal series representations of SL(n, C). The same proof gives the irreducibility of unitary principal series for GL(n) over any local field. Using the fact that the unitary principal series have non-trivial Whittaker models for GL(n), and the uniqueness of the model proved by Rodier ((18)), Howe and Silberger proved in (10) that the unitary principal series of GL(n, F) restricted to SL(n, F) are multiplicity free. The same idea appears in Labesse and Langlands paper (14). In this way, Howe and Silberger obtained that unitary principal series representations of SL(n, F) are multiplicity free. Shahidi observed in (20) that one can prove, using the same idea of Whittaker models, that any irreducible tempered representation of GL(n, F) restricted to SL(n , F) is multiplicity free. In this way one obtains that the parabolically induced representation of SL(«, F) by irreducible tem- pered representation is multiplicity free. A general approach to the reducibility and the multiplicities was done by Keys. The structure of the commuting algebras of unitary principal series representations for Chevalley groups was described by him in (11) and it turned out the multiplicities are not always one. This was also shown earlier by Knapp and Zuckerman in (12). Gelbart and Knapp gave in (5) a de- scription of irreducible constituents of the restriction to SL(w, F) of the unitary principal series representations of GL(n, F). Their pa- per (6) is based on two working hypotheses, the second of them is the multiplicity one of the restriction to SL(«,/ Γ) of irreducible represen- tations of GL(/ι, F). Bernstein showed in (1) that any parabolically

Determinantal method and the Littlewood-Richardson rule

Determinantal method is a powerful tool to study invariants and representations of classical groups. Let R, = Z![x,,,],, , = 1, 2 ,,,,, n be the integral coordinate ring of the module of all n x n integral matrices. In [7], Doubilet, Rota, and Stein gave a beautiful theory on combinatorial approach to invariant theory using a nice basis of R, called standard basis whose elements are simply parametrized by means of “standard tableaux.” This standard basis also occurs in the theory of flag manifolds (cf. [12], [17]). As application of this theory, Procesi and DeConcini gave many results on the characteristic free approach to the invariants and represen- tation spaces of classical groups [4], [S J, 161. In particular, a representation space of each finite dimensional irreducible representation of GL(n, C) is realized in R, @I C canonically. This representation space plays a key roll in application to physics, since each weight vector in this representation space is actually an element of standard basis. Now let us consider the tensor product representation of two arbitrarily irreducible representations pi, and pfl of GL(n, @). Tensor product representations have deep relation with the interaction of particles (cf. [Z]). The branching rule of the tensor product representation pA @ pp into its irreducible constituents is described using combinatorial methods on Young diagrams. This rule, the Littlewood-Richardson rule, was first found by Littlewood [14] and proved rather recently (cf. [15]). However, in application to physics, not only the branching rule but also more detailed information of the representation spaces of tensor product representations is required. In case of GL(2, C), the decomposition of pj.@pp is multiplicity free. So in this case, each weight vector of each irreducible constituent of

Vanishing theorems for tensor powers of an ample vector bundle

LetE be a holomorphic vector bundle of rankr on a compact complex manifoldX of dimensionn. It is shown that the cohomology groupsHp,q(X, E⊗k⊗(detE)l) vanish ifE is ample andp+q≧n+1,l≧n−p+r−1. The proof rests on the well-known fact that every tensor powerE⊗k, splits into irreducible representations of Gl(E). By Bott's theory, each component is canonically isomorphic to the direct image onX of a homogeneous line bundle over a flag manifold ofE. The proof is then reduced to the Kodaira-Akizuki-Nakano vanishing theorem for line bundles by means of the Leray spectral sequence, using backward induction onp. We also obtain a generalization of Le Potier's isomorphism theorem and a counterexample to a vanishing conjecture of Sommese.

On supercuspidal representations of the metaplectic group

The Weil representations associated to anisotropic quadratic forms in one and three variables are used to study supercuspidal representations of the two-fold metaplectic covering group GL ¯ 2 ( k ) {\overline {{\text {GL}}} _2}(k) , where k k is a local nonarchimedean field of odd residual characteristic. The principal result is the explicit calculation of certain Whittaker functionals for any square-integrable irreducible admissible genuine representation of GL ¯ 2 ( k ) {\overline {{\text {GL}}} _2}(k) . In particular, a recent conjecture of Gelbart and Piatetski-Shapiro is answered by obtaining a bijection between the set of quasicharacters of k ∗ {k^ \ast } and the set of irreducible admissible genuine distinguished representations of GL ¯ 2 ( k ) {\overline {{\text {GL}}} _2}(k) , i.e. those representations possessing only one Whittaker functional, or, equivalently, those having a unique Whittaker model. The distinguished representations are precisely the representations attached to the Weil representation associated to a one dimensional form. The local piece of the generalized Shimura correspondence between automorphic forms of GL ¯ 2 ( A ) {\overline {{\text {GL}}} _2}({\mathbf {A}}) and G L 2 ( A ) {\text {G}}{{\text {L}}_2}({\mathbf {A}}) is also treated. Based upon a conjecture of the equivalences among the constituents of the Weil representations associated to two nonequivalent ternary forms, evidence for the explicit form of the local piece of this global correspondence, restricted to supercuspidal representations of GL ¯ 2 ( k ) {\overline {{\text {GL}}} _2}(k) , is presented. In this form, the map is shown to be injective and its image is described.

On The Number of Classes of a Finite Group Invariant for Certain Substitutions

In this paper we consider representations of groups over the field of the complex numbers.The nth-Kronecker power σ⊗n of an irreducible representation σ of a group can be decomposed into the constituents of definite symmetry with respect to the symmetric group Sn. In the special case of the general linear group GL(N) in N dimensions the decomposition of the defining representation at once provides irreducible representations of GL(N) [9; 10; 11].

Весовые диаграммы для пятимерной полной линейной группы

Weight diagrams for irreducible representations of the Lie Algebra of the general linear group in five dimensions over the complex field are classified according to the shape of their boundaries. Weight multiplicities are examined and the reduction under the orthogonal groupO(5, C) of certain homogeneous integral irreducible representations ofGL(5, C) is deduced.