Representations Of Symmetric Group Research Articles

Representations of the symmetric group generated by projected spin functions: A graphical approach

Representations of the symmetric group generated by projected spin functions: A graphical approach

Branching rules for modular representations of symmetric groups, II.

Branching rules for modular representations of symmetric groups, II.

On restrictions of Irreducible Modular Representations of Semisimple Algebraic Groups and Symmetric Groups to Some Natural Subgroups, I

(1994). On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, II. Communications in Algebra: Vol. 22, No. 15, pp. 6175-6208.

On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, II

On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, II

On the black-bodyg-UONIC radiation

Some thermodynamical properties of the black-body radiation are calculated, in which the particles possess a state vector with both the symmetric and antisymmetric representations of the symmetric group.

Representations of symmetric groups with maximal dimension

Results of numerical modeling connected with the study of the asymptotics of representations of the symmetric group SN for largeN which are of maximal dimension are given.

The projective representations of the hyperoctahedral group

The hyperoctahedral group (the Weyl group of the root system B n ) has seven distinct nonsplit double covers (for n ⩾ 4), and hence, seven families of projective representations. We give constructions of all the irreducible representations of these seven double covers in terms of symmetric group representations, and determine the associated character tables. As a corollary, we also obtain the irreducible projective representations and characters of the Weyl group of the root system D n .

HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND TENSOR PRODUCTS OF SYMMETRIC GROUP REPRESENTATIONS

The Hopf algebra structure of the ring of symmetric functions is used to prove a new identity for the internal product, i.e., the operation corresponding to the tensor product of symmetric group representations. From this identity, or by similar techniques which can also involve the λ-ring structure, we derive easy proofs of most known results about this operation. Some of these results are generalized.

Multivariate polynomials, standard tableaux, and representations of symmetric groups

This paper is concerned with structural and algorithmic aspects of certain R-bases in polynomial rings R[ X ij ] over a commutative ring R with 1. These bases are related to standard tableaux. We shall examine the main tools in full detail: (symmetrized) bideterminants, Capelli operators, hyperdominance, and generalized Laplace's expansions. These tools are then applied to the representation theory of symmetric groups. In particular, we present an algorithm which efficiently computes for every skew module of a symmetric group an R-basis which is adapted to a Specht series. This result is a constructive, characteristic-free analogue of the celebrated Littlewood-Richardson rule. This paper will serve as the basis for a possible generalization of that rule to more general shapes.

Generalized commutation relations for a single-mode oscillator.

Recently Greenberg1 constructed an example of infinite statistics in which all the representations of the symmetric group can occur and the number operator is an infinite series in a and a †. In this paper, we show that we can write down a generalized commutation relation (CR), for a single mode for which the number operators is again an infinite series and contains BE, FD and Greenberg’s infinite statistics (IS) as special cases. We discuss the quantization of free fields with infinite statistics following the procedure of Umezawa and Takahashi.2

Mapping class groups of low genus and their cohomology

Concerned with the calculation of the cohomology of the mapping class group of a closed oriented surface of genus two, this book uses methods involving braid groups, modular representations of symmetric groups and configuration spaces.

Weights for symmetric and general linear groups

An important feature of the theory of finite groups is the number of connections and analogies with the theory of Lie groups. The concept of a weight has long been useful in the modular representation theory of finite Lie groups in the defining characteristic of the group. The idea of a weight in the modular representation theory of an arbitrary finite group was recently introduced in Alperin ( Proc. Sympos. Pure Math. 41 (1987, 369–379), where it was conjectured that the number of weights should equal the number of modular irreducible representations. Moreover, this equality should hold block by block. The conjecture has created great interest, since its truth would have important consequences—a synthesis of known results and solutions of outstanding problems. In this paper we prove the conjecture first for the modular representations of symmetric groups and second for modular representations in odd characteristic r for the finite general linear groups. In the latter case r may be assumed to be different from the defining characteristic p of the group, since the result is known when r is p. The well-known analogy between the representation theory of the symmetric and general linear groups holds here too.

Spread-invariant representations of symmetric groups

Spread-invariant representations of symmetric groups

Explicit formulas for the Hamiltonian matrix elements

AbstractThe explicit formulas for the evaluation of the Hamiltonian matrix elements are presented. The calculation of the integral coefficients is independent of both the nature of the orbitals and th spin coupling schemes. It is fully automatic and only dependent on the number of doubly and singly occupied orbitals. Further‐more, the symmetric group representation matrices are not needed, and the N! problem can be avoided.

Representations of symmetric groups by bad shapes

Representations of symmetric groups by bad shapes

On Schur algebras and related algebras, II

This paper is a continuation of [7]. The purpose of [7] was to set up, in the context of reductive algebraic groups, a theory of “generalized Schur algebras.” A special case of our construction gives the Schur algebras S(n, r) studied extensively in [lo]. We concentrate here on the Schur algebras themselves and show that our construction yields new results in their representation theory. By means of the Schur functor (see [ 10, Chap. 6]), we obtain new results in the representation theory of the symmetric groups. The main results are as follows. The Schur algebras S,(n, r), defined over a principal ideal domain R, have finite global dimension (see also [ 1 I). Over a field or complete valuation ring R, every finitely generated projective S,(n, r)-module has a Weyl module filtration. We prove the irreducibility of the socle of certain Weyl modules (see also [15]). We describe a connection between weights of projective indecomposable modules for the Schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation. We show that, over a complete valuation ring or a field, every Young module for a symmetric group has a Specht series, in particular the projective indecomposables have Specht series. Finally, we show that, for r <n and a complete valuation ring or field R, the blocks of the Schur algebra S,(n, r) are given by the rule embodied in Nakayama’s conjecture. The notations and conventions are as in the first part of this work [7].

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.

Tensors of symmetric and unitary groups

A realization of unitary-group basis states by multinomials (tensors) that realize symmetric-group representation basis states requires that there be the proper set of numbers of these tensors and that unitary-group transformation properties be obtainable from their labels. That these requirements are met is shown here, allowing the actual construction of the unitary states and matrix elements.

W -graphs of representations of symmetric groups

W -graphs are closely connected with Kazhdan-Lustig polynomials, arising in connection with problems of the theory of representations of semisimple groups. For the symmetric group Sh a description is known of the Kazhdan-Lustig polynomials and the W -graphs connected with them for the Grassman manifolds. The goal of this paper is a combinatorial description of more general graphs. In particular, the graphs of hooks and all irreducible W -graphs for n=6 are found.

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 Sn 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 η-function.