Ordinary Irreducible Representations Research Articles

On the irreducible spin representations of symmetric and alternating groups which remain irreducible in characteristic 3

For any finite group G G and any prime p p one can ask which ordinary irreducible representations remain irreducible in characteristic p p , or more generally, which representations remain homogeneous in characteristic p p . In this paper we address this question when G G is a proper double cover of the symmetric or alternating group. We obtain a classification when p = 3 p=3 except in the case of a certain family of partitions relating to spin RoCK blocks. Our techniques involve induction and restriction, degree calculations, decomposing projective characters and recent results of Kleshchev and Livesey on spin RoCK blocks.

The minimal polynomials of powers of cycles in the ordinary representations of symmetric and alternating groups

Denote the alternating and symmetric groups of degree [Formula: see text] by [Formula: see text] and [Formula: see text], respectively. Consider a permutation [Formula: see text], all of whose nontrivial cycles are of the same length. We find the minimal polynomials of [Formula: see text] in the ordinary irreducible representations of [Formula: see text] and [Formula: see text].

Irreducible projective representations of the alternating group which remain irreducible in characteristic 2

For any finite group G it is an interesting question to ask which ordinary irreducible representations of G remain irreducible in a given characteristic p. We answer this question for p=2 when G is the proper double cover of the alternating group. As a key ingredient in the proof, we prove a formula for the decomposition numbers in Rouquier blocks of double covers of symmetric groups, in terms of Schur P-functions.

Irreducible projective representations of the symmetric group which remain irreducible in characteristic 2

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$. We answer this question for $p=2$ when $G$ is a proper double cover of the symmetric group. Our techniques involve constructing part of the decomposition matrix for a Rouquier block of a double cover, restricting to subgroups using the Brundan--Kleshchev modular branching rules and comparing the dimensions of irreducible representations via the bar-length formula.

P-Saturations of Welter’s game and the irreducible representations of symmetric groups

We establish a relation between the Sprague-Grundy function $\text{sg}$ of a $p$-saturation of Welter's game and the degrees of the ordinary irreducible representations of symmetric groups. In this game, a position can be viewed as a partition $\lambda$. Let $\rho^\lambda$ be the irreducible representation of $\text{Sym}(|\lambda|)$ indexed by $\lambda$. For every prime $p$, we show the following results: (1) the degree of $\rho^\lambda$ is prime to $p$ if and only if $\text{sg}(\lambda) = |\lambda|$; (2) the restriction of $\rho^\lambda$ to $\text{Sym}(\text{sg}(\lambda))$ has an irreducible component with degree prime to $p$. Further, for every integer $p$ greater than 1, we obtain an explicit formula for $\text{sg}(\lambda)$.

The irreducible representations of the alternating group which remain irreducible in characteristic 𝑝

Let p p be an odd prime, and A n \mathfrak {A}_n the alternating group of degree n n . We determine which ordinary irreducible representations of A n \mathfrak {A}_n remain irreducible in characteristic p p , verifying the author’s conjecture from 2010. Given the preparatory work done in 2010, our task is to determine which self-conjugate partitions label Specht modules for the symmetric group in characteristic p p having exactly two composition factors. This is accomplished through the use of the Robinson–Brundan–Kleshchev ‘ i i -restriction’ functors, together with known results on decomposition numbers for the symmetric group and additional results on the Mullineux map and homomorphisms between Specht modules.

On the irreducible representations of the alternating group which remain irreducible in characteristic 𝑝

We consider the problem of which ordinary irreducible representations of the alternating group A n \mathfrak {A}_n remain irreducible modulo a prime p p . We solve this problem for p = 2 p=2 , and present a conjecture for odd p p , which we prove in one direction.

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 j i . 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 [λ] , 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.

Generating fast Fourier transforms of solvable groups

This paper presents a new algorithm for constructing a complete list of pairwise inequivalent ordinary irreducible representations of a finite solvable group G. The input of the algorithm is a pc presentation corresponding to a composition series refining a chief series of G. Modifying the Baum–Clausen algorithm for supersolvable groups and combining this with an idea of Plesken for constructing intertwining spaces, we derive a worst-case upper complexity bound O( p·| G| 2log(| G|)), where p is the largest prime divisor of | G|. The output of the algorithm is well suited to performing a fast Fourier transform of G. For supersolvable groups there are composition series which are already chief series. In this case the generation of discrete Fourier transforms can be done more efficiently than in the solvable case. We report on a recent implementation for the class of supersolvable groups.

On Specht modules of general linear groups

We wish to consider the reduction modulo p of the ordinary irreducible unipotent representations of the general linear groups GLn(q) in the case that p q . For each partition λ of n, James [3] defined a GLn(q)-module S, known as a Specht module. Over fields of characteristic zero, these Specht modules form the complete set of nonisomorphic irreducible unipotent GLn(q)-modules. They are usually defined in terms of the intersection of the kernels of certain homomorphisms, and a generating element can be found for each Specht module. However, in all but a few cases, no explicit basis for S as a vector space has been found. The results presented here demonstrate a property of the Specht modules which may help to determine their bases; it severely restricts the form of any such basis vector.

Some reducible Specht modules

We wish to consider which ordinary irreducible representations of the symmetric group Sn remain irreducible modulo a prime p; this is the same as asking which partitions λ of n have the property that the corresponding Specht module S is reducible over a field of characteristic p. If λ or its conjugate partition λ′ is p-regular then the answer is known [6,8]; it is also known in the case p = 2 [7]. This paper discusses some of the reducible Specht modules in the case that p 3. Throughout,p will be an odd prime and λ a partition of some integer n. We will prove some cases of a conjecture by James and Mathas [9], given below. We begin with some definitions.

The Irreducible Specht Modules in Characteristic 2

In the representation theory of finite groups, it is useful to know which ordinary irreducible representations remain irreducible modulo a prime p. For the symmetric groups Sn, this amounts to determining which Specht modules are irreducible over a field of characteristic p. Throughout this note we work in characteristic 2, and in this case we classify the irreducible Specht modules, thereby verifying the conjecture in [3, p. 97]. Recall that a partition is 2-regular if all of its non-zero parts are distinct; otherwise the partition is 2-singular. The irreducible Specht modules Sλ with λ a 2-regular partition were classified in [2]. Let λ′ denote the partition conjugate to λ. If Sλ is irreducible, then Sλ′ is irreducible, since Sλ′ is isomorphic to the dual of Sλ tensored with the sign representation. It turns out that if neither λ nor λ′ is 2-regular, then Sλ is irreducible only if λ = (2, 2). In order to state our theorem, we let l(k), for an integer k, be the least non-negative integer such that k < 2l(k). 1991 Mathematics Subject Classification 20C30.

Minimal set of class-sums characterizing the ordinary irreducible representations of the symmetric group, and the Tarry-Escott problem

The shape of a Young diagram Y (| Y| = n) can be specified in terms of the set of symmetric power sums over its contents, σ l = Σ ( ij) ϵY ( j − i) l ; l = 1, 2, …, n. It is remarkable that the set of power sums σ 1, σ 2, …, σ k is sufficient to characterize the Young diagrams possessing up to n( k) boxes, where n( k) is considerably larger than k. Numerical evidence for k⩽5 is roughly consistent with n(k) ∼ 4( 4 3 ) 2k . The lower bound n( k) > k + max(2, √ k) has been derived by examination of some properties of Young diagrams with a large number of rows, and the upper bound n( k) < 2 2 k+1 has been established using a variant of the Tarry-Escott problem.

Computing irreducible representations of supersolvable groups

Recently, it has been shown that the ordinary irreducible representations of a supersolvable group G of order n given by a power-commutator presentation can be constructed in time O ( n 2 log ⁡ n ) O({n^2}\log n) . We present an improved algorithm with running time O ( n log ⁡ n ) O(n\log n) .

A new look at the conformal invariance in QFT

Using a infinite-dimensional, nondecomposable representation of the dilatation subgroup, it is shown that, in the general case, β≠0 the Calan-Symanzik equation coincides with the naive scaling law. It is pointed out that this representation leaves the perturbative series invariant. In the limiting case β=0, this representation degenerates to the ordinary irreducible representation of the dilatation group.

The evaluation of irreducible polynomial representations of the general linear groups and of the unitary groups over fields of characteristic 0

We describe an efficient method for the computer evaluation of the ordinary irreducible polynomial representations of general linear groups using an integral form of the ordinary irreducible representations of symmetric groups. In order to do this, we first give an algebraic explanation of D. E. Littlewood's modification of I. Schur's construction. Then we derive a formula for the entries of the representing matrices which is much more concise and adapted to the effective use of computer calculations. Finally, we describe how one obtains — using this time an orthogonal form of the ordinary irreducible representations of symmetric groups — a version which yields a unitary representation when it is restricted to the unitary subgroup. In this way we adapt D. B. Hunter's results which heavily rely on Littlewood's methods, and boson polynomials come into the play so that we also meet the needs of applications to physics.

Character constraints on duality

Duality identifies recoupling coefficients and isoscalar factors of the unitary groups with matrix elements of double coset representatives of the symmetric groups. When the double coset factorization is with respect to the same subgroup chain the matrix is that of the ordinary irreducible representation in a basis symmetry adapted to the subgroup chain. The invariant character of the symmetric group element then acts as a constraint on the magnitude, phase, and multiplicity resolution for diagonal elements. An algorithm for determining the content of this constraint is given and is shown to be consistent with a phase convention proposed previously.

On the minimal dimensions of irreducible representations of symmetric groups

For each integer m, Rasala [6] has shown how to list all the ordinary irreducible representations of the symmetric group n which have degree less than nm, provided that n is large enough, and in this note we shall prove similar results for the irreducible representations of n over an arbitrary field K. Our estimates are very crude, so although we recover Rasala's results, we get nowhere near his precise information on how large n has to be.

Some combinatorial results involving Young diagrams

In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.

On a conjecture of Carter concerning irreducible Specht modules

We study the question: Which ordinary irreducible representations of the symmetric group remain irreducible modulo a prime p?Let Sλ be the Specht module corresponding to the partition λ of n. The definition of Sλ is ‘independent of the field we are working over’. When the field has characteristic zero, Sλ is irreducible, and gives the ordinary irreducible representation of corresponding to the partition λ. Thus we are interested in the problem of whether or not Sλ is irreducible over a field of characteristic p.