Modular irreducible representations of the symmetric group as linear codes

We describe a particularly easy way of evaluating the modular irreducible matrix representations of the symmetric group. It shows that Specht’s approach to the ordinary irreducible representations, along Specht polynomials, can be unified with Clausen’s approach to the modular irreducible representations using symmetrized standard bideterminants. The unified method, using symmetrized Specht polynomials, is very easy to explain, and it follows directly from Clausen’s theorem by replacing the indeterminate x_ij of the letter place algebra by x_jⁱ.Our approach is implemented in SYMMETRICA. It was used in order to obtain computational results on code theoretic properties of the p-modular irreducible representation [λ]_p corresponding to a p-regular partition λ via embedding it into representation spaces obtained from ordinary irreducible representations. The first embedding is into the permutation representation induced from the column group of a standard Young tableau of shape λ. The second embedding is the embedding of [λ]_p into the space of , the p-modular representation obtained from the ordinary irreducible representation [λ] by reducing the coefficients modulo p.We include a few tables with dimensions and minimum distances of these codes; others can be found via our home page.     

On Thompson's Theorem

Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ₁ and χ₂ of distinct degrees such that p does not divide χ₁(1)χ₂(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed.     

Theory of path length distributions I

The Path Length Distribution (PLD) of a (p, q) graph is defined to be the array (X₀, X₁, X₂, …, X_p-1), where X₀ is the number of (unordered) pairs of vertices which have no path connecting them and X_l, 1 ≦ l ≦ p-1, is the number of pairs of vertices which are connected by a path of length l (see [1, 2]). The topic of this paper is the occurence of non-isomorphic graphs having the same path length distribution. For trees, a constructive procedure is given, showing that for any positive integer N there exist N non-isomorphic trees of diameter four which have the same PLD. Also considered are PLD-maximal graphs — those graphs with p vertices such that all pairs of vertices are connected by a path of length l for 2 ≦ l ≦ p-1. In addition to providing more examples of non-isomorphic graphs having the same PLD, PLD-maximal graphs are of intrinsic interest. For PLD-maximal graphs, we give sufficient degree and edge conditions and a necessary edge condition.     

Globally Irreducible Representations of Finite Groups and Integral Lattices

The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L ₂(q) and ²B₂(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pⁿ-1)/2 of the symplectic group Sp_2n(p) (p 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pⁿ–1). A summary of currently known globally irreducible representations is given.     

Representations of Lie superalgebras in prime characteristic II: The queer series

The modular representation theory of the queer Lie superalgebra q(n) over characteristic p>2 is developed. We obtain a criterion for the irreducibility of baby Verma modules with semisimple p-characters χ and a criterion for the semisimplicity of the corresponding reduced enveloping algebras U_χ(q(n)). A (2p)-power divisibility of dimensions of q(n)-modules with nilpotent p-characters is established. The representation theory of q(2) is treated in detail. We formulate a Morita super-equivalence conjecture for q(n) with general p-characters which is verified for n=2.     

Subdirectly Irreducible Double Demi-p-Lattices

The aim of this paper is to give a characterization of the (finitely) subdirectly irreducible double demi-p-lattices. First, we prove a congruence representation theorem for double demi-p-lattices, which is a natural analogue of the theorem given in [2] for double p-algebras. These results are inspired by the representation theorem given by Lakser [6] for p-algebras, and yield a natural approach to the study of subdirectly irreducible algebras.     

Finite groups with a <Emphasis Type="Italic">D</Emphasis>-block of cardinality 3

The results of this paper are connected with one of the main problems of the representation theory, the problem of dependence between the character table of a finite group and its abstract structure. Finite groups that have a D-block of cardinality 3 for some normal subset D are investigated. Some properties of classical p-blocks (where p is a prime number) consisting of three irreducible characters are also obtained.     

Distribution of Wilks' likelihood-ratio criterion in the complex case

Summary The null distribution of Wilks' likelihood ratio criterian, Λ, in the complex case, is obtained, and explicit expressions for the same are given forp=2 and 3, wherep is the number of variables. It is shown that unlike the real case the distributions derived have closed form representation for allp and for allf ₂, the hypothesis degree of freedom. Tables of correction factors for converting chi-square percentiles to exact percentiles of a logarithmic function of Λ are provided for fourteen (p, f ₂) pairs. Tables for an additional thirteen pairs can be obtained from those tabulated by interchangingp andf ₂. This research was supported (in part) by the National Science Foundation under Grant Number GU-15-34. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equations on

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q=2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Möbius group of conformal transformations on , and preserves a space of solutions of the ultrahyperbolic Laplace equation on . We construct in an intrinsic and natural way a Hilbert space of solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on L²(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p,q).     

Two-dimensional l-adic representations of the Galois group of a global field of characteristic P and automorphic forms on Gl(2)

It is well known (Ref. Zh. Mat., 1978, 1A405) that to each parabolic representation of the group GL (2) over adeles of a global field k of characteristic p there corresponds an irreducible two-dimensionall-adic representation of the Galois group of this field. In this paper, it is proved that, conversely also, to each irreducible two-dimensionall-adic representation of the Galois group there corresponds a parabolic representation of the group GL(2) over adeles. The proof of Langland's hypotheses for GL(2,k) is thereby completed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 138–156, 1984.     

A basis for the symplectic group branching algebra

The symplectic group branching algebra, B\mathcal {B}, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp_2n−2(ℂ) in each finite-dimensional irreducible representation of Sp_2n (ℂ). By describing on B\mathcal {B} an ASL structure, we construct an explicit standard monomial basis of B\mathcal {B} consisting of Sp_2n−2(ℂ) highest weight vectors. Moreover, B\mathcal {B} is known to carry a canonical action of the n-fold product SL₂×⋯×SL₂, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi algebras we describe a deformation of Spec(B)\mathrm{Spec}(\mathcal {B}) into an explicitly described toric variety.     

Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equations on

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q=2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Möbius group of conformal transformations on , and preserves a space of solutions of the ultrahyperbolic Laplace equation on . We construct in an intrinsic and natural way a Hilbert space of solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on L²(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p,q).     

Steiner systems with automorphism groups PSL(2, 71), PSL(2, 83), and P∑L(2, 35)

The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were found by Mills [10]. We presently enumerate all 5-(72, 6, 1) designs with this automorphism group. There are in all 926299 non-isomorphic designs. We show that a necessary condition for semiregular5-(v, 6, 1) designs with automorphism group PSL(2, v 1) to exist is thatv=84, 228 (mod 360). In particular, there are exactly 3 non-isomorphic semiregular 5-(84, 6, 1) designs with automorphism group PSL(2, 83). There are at least 6450 non-isomorphic 5-(244, 6, 1) designs with automorphism group PL(2, 3⁵).     

Geometric interpretations of two branching theorems of D.E. Littlewood

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τ_λ of U(n) has nonnegative highest weight λ of depth n/2. It says that the multiplicity in τ_λ|_SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τ_λ in the U(n) tensor product τ_μτ_ν, the allowable μ's being all even nonnegative highest weights for U(n). Littlewood's proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τ_μτ_ν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.     

The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.     

Free group representations from vector-valued multiplicative functions,I

Let Γ denote a noncommutative free group and let Ω stand for its boundary. We construct a large class of unitary representations of Γ. This class contains many previously studied representations, and is closed under several natural operations. Each of the constructed representations is in fact a representation of Γ ⋉_λ C(Ω). We prove here that each of them is irreducible as a representation of Γ ⋉_λ C(Ω). Actually, as will be shown in further work, each of them is irreducible as a representation of Γ, or is the direct sum of exactly two irreducible, inequivalent Γ-representations. This research was supported by the Italian CNR.     

An irreducible smooth non-admissible representation

It is shown for the group of k-rational points of an affine algebraic group G with k a finite extension of Q_p that the topological irreducibility of unitary representations of G does not have to be equivalent to the algebraic irreducibility of the representation on the smooth vectors. We give for a specific G an example of an irreducible smooth representation, which is not admissible.     

Expected Reflection Distance in G(r, 1, n) after a Fixed Number of Reflections

Extending to r > 1 a formula of the authors, we compute the expected reflection distance of a product of t random reflections in the complex reflection group G(r, 1, n). The result relies on an explicit decomposition of the reflection distance function into irreducible G(r, 1, n)-characters and on the eigenvalues of certain adjacency matrices.Received December 8, 2003     

On the tensor product of a finite and an infinite dimensional representation

There are two main results in the paper. The first gives the infinitesimal character that can occur in the tensor product V V_λ of an irreducible finite dimensional representation V_λ and an irreducible infinite dimensional representation V of a semisimple Lie algebra . The statement is that the infinitesimal characters are x_{v + μi}, I = 1, 2,…, k, where μ_i are the weights of V_λand v is the “pseudo” highest weight of V.The second result proves that if V is a Harish-Chandra module (one which comes from a group representation), then V V_λ has a finite composition series. But then the irreducible components in the composition series have the infinitesimal characters given in the first results.