Irreducible Representations Of Symmetric Group Research Articles

A web basis of invariant polynomials from noncrossing partitions

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules Sλ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases λ=(d,d) and λ=(d,d,d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d,d,1ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.

Stability phenomena for resonance arrangements

We prove that the i th i^\text {th} graded pieces of the Orlik–Solomon algebras or Cordovil algebras of resonance arrangements form a finitely generated FS o p \operatorname {FS}^{\mathrm {op}} -module, thus obtaining information about the growth of their dimensions and restrictions on the irreducible representations of symmetric groups that they contain.

Lower bounds for dimensions of irreducible representations of symmetric groups

We give new, explicit and asymptotically sharp, lower bounds for dimensions of irreducible modular representations of finite symmetric groups.

Horn inequalities for nonzero Kronecker coefficients

The Kronecker coefficients gαβγ and the Littlewood- Richardson coefficients cαβγ are nonnegative integers depending on three partitions α, β, and γ. By definition, gαβγ (resp. cαβγ) are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups (resp. linear groups). By a classical Littlewood-Murnaghan's result the Kronecker coefficients extend the Littlewood-Richardson ones.The nonvanishing of the Littlewood-Richardson coefficient cαβγ implies that (α,β,γ) satisfies some linear inequalities called Horn inequalities. In this paper, we extend the essential Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient.Along the way, we describe the set of tripless (α,β,γ) of partitions such that cαβγ≠0 and l(α)≤e, l(β)≤f and l(γ)≤e+f, for some given positive integers e and f. This set is the natural analogue of the classical Horn semigroup when one thinks about cαβγ as the branching multiplicities for the subgroup GLe×GLf of GLe+f.

Irreducible modules for symmetric groups that are summands of their exterior square

A group representation is said to have the ESQ property if it is isomorphic to a quotient of its own exterior square. In the present paper we will investigate the ESQ property for some particular irreducible representations of symmetric groups over the complex field. We consider the representations corresponding to height two, width two, and hook shape Young tableaux.

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 $\text{sg}(\lambda)$.

Entanglement classification in the noninteracting Fermi gas

In this paper, entanglement classification shared among the spins of localized fermions in the noninteracting Fermi gas is studied. It is proven that the Fermi gas density matrix is block diagonal on the basis of the projection operators to the irreducible representations of symmetric group [Formula: see text]. Every block of density matrix is in the form of the direct product of a matrix and identity matrix. Then it is useful to study entanglement in every block of density matrix separately. The basis of corresponding Hilbert space are identified from the Schur–Weyl duality theorem. Also, it can be shown that the symmetric part of the density matrix is fully separable. Then it has been shown that the entanglement measure which is introduced in Eltschka et al. [New J. Phys. 10, 043104 (2008)] and Guhne et al. [New J. Phys. 7, 229 (2005)], is zero for the even [Formula: see text] qubit Fermi gas density matrix. Then by focusing on three spin reduced density matrix, the entanglement classes have been investigated. In three qubit states there is an entanglement measure which is called 3-tangle. It can be shown that 3-tangle is zero for three qubit density matrix, but the density matrix is not biseparable for all possible values of its parameters and its eigenvectors are in the form of W-states. Then an entanglement witness for detecting non-separable state and an entanglement witness for detecting nonbiseparable states, have been introduced for three qubit density matrix by using convex optimization problem. Finally, the four spin reduced density matrix has been investigated by restricting the density matrix to the irreducible representations of [Formula: see text]. The restricted density matrix to the subspaces of the irreducible representations: [Formula: see text], [Formula: see text] and [Formula: see text] are denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. It has been shown that some highly entangled classes (by using the results of Miyake [Phys. Rev. A 67, 012108 (2003)] for entanglement classification) do not exist in the blocks of density matrix [Formula: see text] and [Formula: see text], so these classes do not exist in the total Fermi gas density matrix.

On the asymptotics of Kronecker coefficients

Kronecker coefficients encode the tensor products of complex irreducible representations of symmetric groups. Their stability properties have been considered recently by several authors (Vallejo, Pak and Panova, Stembridge). We describe a geometric method, based on Schur---Weyl duality, that allows to produce huge series of instances of this phenomenon. Moreover, the method gives access to lots of extra information. Most notably, we can often compute the stable Kronecker coefficients, sometimes as numbers of points in very explicit polytopes. We can also describe explicitly the moment polytope in the neighbourhood of our stable triples. Finally, we explain an observation of Stembridge on the behaviour of certain rectangular Kronecker coefficients, by relating it to the affine Dynkin diagram of type $$E_6$$E6.

Asymptotics of characters of symmetric groups, genus expansion and free probability

The convolution of indicators of two conjugacy classes on the symmetric group S q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys–Murphy element involves many conjugacy classes with complicated coefficients. In this article, we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a two-dimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles the genus expansion from the random matrix theory. As the main application we study irreducible representations of symmetric groups S q for large q. We find the asymptotic behavior of characters when the corresponding Young diagram rescaled by a factor q - 1 / 2 converge to a prescribed shape. The character formula (known as the Kerov polynomial) can be viewed as a power series, the terms of which correspond to two-dimensional surfaces with prescribed genus and we compute explicitly the first two terms, thus we prove a conjecture of Biane.

Specht modules and chromatic polynomials

An explicit formula for the chromatic polynomials of certain families of graphs, called `bracelets', is obtained. The terms correspond to irreducible representations of symmetric groups. The theory is developed using the standard bases for the Specht modules of representation theory, and leads to an effective means of calculation.

The finite basis property for some classes of irreducible representations of symmetric groups

The possibility of defining irreducible representations of symmetric groups by means of finitely many relations is studied. The existence of finite bases is established for the classes of representations corresponding to two-part partitions and to partitions in the fundamental alcove.

Kawanaka Invariants for Representations of Weyl Groups

Let W be a Weyl group and let V be the natural CW-module, i.e., the reflection representation. For a complex irreducible character χ of W, we consider the invariantIχ;q≔|W|−1∑w∈Wχw2det1+qw|V/det1−qw|Vintroduced by N. Kawanaka. We determine I(χ;q) explicitly. Looking over these results, we observe a relation between Kawanaka's invariants I(χ;q) and the two-sided cells. For example, if a two-sided cell consists of a single element χ, then the Kawanaka invariant I(χ;q) can be expressed as ∏li=1(1+qhi)/(1−qhi) with some integers hi. This expression can be regarded as a quantization of the usual hook formula for the dimension of irreducible representations of symmetric groups.

The globally irreducible representations of symmetric groups

Let $K$ be an algebraic number field and $\mathcal {O}$ be the ring of integers of $K$. Let $G$ be a finite group and $M$ be a finitely generated torsion free $\mathcal {O} G$-module. We say that $M$ is a globally irreducible $\mathcal {O} G$-module if, for every maximal ideal $\mathfrak {p}$ of $\mathcal {O}$, the $k_\mathfrak {p} G$-module $M\otimes _{ \mathcal {O}} k_\mathfrak {p}$ is irreducible, where $k_\mathfrak {p}$ stands for the residue field $\mathcal {O}/\mathfrak {p}$. Answering a question of Pham Huu Tiep, we prove that the symmetric group $\Sigma _n$ does not have non-trivial globally irreducible modules. More precisely we establish that if $M$ is a globally irreducible $\mathcal {O} \Sigma _n$-module, then $M$ is an $\mathcal {O}$-module of rank $1$ with the trivial or sign action of $\Sigma _n$.

Lower bound on the dimensions or irreducible representations of symmetric groups and on the exponents of varieties of Lie algebras

Two main results of this paper are singled out. The first one relates to the representation theory of symmetric groups. The second one deals with varieties of Lie algebras over a field of characteristic zero. The first result can be presented as follows: given a symmetric group of sufficiently large degree , every irreducible representation of it with Young diagram fitting into a square with side is of dimension at least . The second result states that there are no varieties of Lie algebras over a field of characteristic zero with lower exponent strictly less than two. At the same time, examples of varieties with exponent two are presented.

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.

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.

Tables of Reductions of Symmetrized Inner Products ("Inner Plethysms") of Ordinary Irreducible Representations of Symmetric Groups

Tables of Reductions of Symmetrized Inner Products ("Inner Plethysms") of Ordinary Irreducible Representations of Symmetric Groups

Tables of reductions of symmetrized inner products (“inner plethysms”) of ordinary irreducible representations of symmetric groups

Decompositions of symmetrized inner products [ α ] ⊡ [ β ] [\alpha ] \boxdot [\beta ] of ordinary irreducible representations [ α ] [\alpha ] of symmetric groups S n {S_n} and [ β ] [\beta ] of S m {S_m} were evaluated on a CDC 6400. Tables were obtained for 2 ⩽ n ⩽ 10 2 \leqslant n \leqslant 10 and 2 ⩽ m ⩽ 5 2 \leqslant m \leqslant 5 as well as for m = 6 m = 6 and 2 ⩽ n ⩽ 7 2 \leqslant n \leqslant 7 .

On the Degrees of the Irreducible Representations of Symmetric Groups

Bulletin of the London Mathematical SocietyVolume 3, Issue 2 p. 189-192 Notes and papers On the Degrees of the Irreducible Representations of Symmetric Groups I. G. MacDonald, I. G. MacDonald Magdalen College, OxfordSearch for more papers by this author I. G. MacDonald, I. G. MacDonald Magdalen College, OxfordSearch for more papers by this author First published: July 1971 https://doi.org/10.1112/blms/3.2.189Citations: 31AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume3, Issue2July 1971Pages 189-192 RelatedInformation