Representations Of Symmetric Group Research Articles

Diagonal free field matrix correlators, global symmetries and giant gravitons

We obtain a basis of diagonal free field multi-matrix 2-point correlators in a theory with global symmetry group G. The operators fall into irreducible representations of G. This applies for gauge group U(N) at finite N. For composites made of n fundamental fields, this is expressed in terms of Clebsch-Gordan coefficients for the decomposition of the n-fold tensor products of the fundamental field representation in terms of G \times S_n representations. We use this general construction in the case of the SL(2) sector of \cN=4 SYM. In this case, by using oscillator constructions, we reduce the computation of the relevant Clebsch-Gordans coupling infinite dimensional discrete series irreps of SL(2) to a problem in symmetric groups. Applying these constructions we write down gauge invariant operators with a Fock space structure similar to that arising in a large angular momentum limit of worldvolume excitations of giant gravitons. The Fock space structure emerges from Clebsch multiplicities of tensor products of symmetric group representations. We also give the action of the 1-loop dilatation operator of \cN=4 SYM on this basis of multi-matrix operators.

Torus quotients of homogeneous spaces of the general linear group and the standard representation of certain symmetric groups

We give a stratification of the GIT quotient of the Grassmannian G2,n modulo the normaliser of a maximal torus of SLn(k) with respect to the ample generator of the Picard group of G2,n. We also prove that the flag variety GLn(k)/Bn can be obtained as a GIT quotient of GLn+1(k)/Bn+1 modulo a maximal torus of SLn+1(k) for a suitable choice of an ample line bundle on GLn+1(k)/Bn+1.

Multiplicity-free representations of symmetric groups

Building on work of Saxl, we classify the multiplicity-free permutation characters of all symmetric groups of degree 66 or more. A corollary is a complete list of the irreducible characters of symmetric groups (again of degree 66 or more) which may appear in a multiplicity-free permutation representation. The multiplicity-free characters in a related family of monomial characters are also classified. We end by investigating a consequence of these results for Specht filtrations of permutation modules defined over fields of prime characteristic.

Parking functions and vertex operators

We introduce several associative algebras and families of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we define a family of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct families of vector spaces whose dimensions are Catalan numbers and Fuss–Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras.

An insertion operator preserving infinite reduction sequences

A common way to show the termination of the union of two abstract reduction systems, provided both systems terminate, is to prove that they enjoy a specific property (some sort of ‘commutation’ for instance). This specific property is actually used to show that, for the union not to terminate, one of the systems must itself be non-terminating, which leads to a contradiction. Unfortunately, the property may be impossible to prove because some of the objects that are reduced do not enjoy an adequate form.Hence the purpose of this paper is threefold:–First, it introduces an operator enabling us to insert a reduction step on such an object, and therefore to change its shape, while still preserving the ability to use the property. Of course, some new properties will need to be verified.–Second, as an instance of our technique, the operator is applied to relax a well-known lemma stating the termination of the union of two termination abstract reduction systems.–Finally, this lemma is applied in a peculiar and then in a more general way to show the termination of some lambda calculi with inductive types augmented with specific reductions dealing with:(i)copies of inductive types;(ii)the representation of symmetric groups.

REPRESENTATIONS OF THE SYMPLECTIC ROOK MONOID

In this paper, we concern representations of symplectic rook monoids R. First, an algebraic description of R as a submonoid of a rook monoid is obtained. Second, we determine irreducible representations of R in terms of the irreducible representations of certain symmetric groups and those of the symplectic Weyl group W. We then give the character formula of R using the character of W and that of the symmetric groups. A practical algorithm is provided to make the formula user-friendly. At last we show that the Munn character table of R is a block upper triangular matrix.

K. Saito's Conjecture for nonnegative eta products and analogous results for other infinite products

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z = 1 case is an identity for the generating function for p-cores due to Klyachko [A.A. Klyachko, Modular forms and representations of symmetric groups, J. Soviet Math. 26 (1984) 1879–1887] and Garvan, Kim and Stanton [F. Garvan, D. Kim, D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990) 1–17]. A number of other infinite products are shown to have nonnegative coefficients. In the process a new generalization of the quintuple product identity is derived.

Q-Analogues of regularisation theorems for linear and projective representations of the symmetric group

Brundan and Kleshchev have recently proved an analogue of James's ‘regularisation theorem’ for modular representations of symmetric groups, in the context of projective representations. We prove ‘ q-analogues’ of both theorems, i.e. corresponding theorems concerning canonical basis coefficients for basic representations of quantum affine algebras of types A e − 1 ( 1 ) and A 2 n ( 2 ) .

Markov Measures on Young Tableaux and Induced Representations of the Infinite Symmetric Group

We show that the class of so‐called Markov representations of the infinite symmetric group ${\mathfrak G}_{\bf N}$, associated with Markov measures on the space of infinite Young tableaux, coincides with the class of simple representations, i.e., inductive limits of representations with simple spectrum. The spectral measure of an arbitrary representation of ${\mathfrak G}_{\bf N}$ with simple spectrum is equivalent to a multi‐Markov measure on the space of Young tableaux. We also show that the representations of ${\mathfrak G}_{\bf N}$ induced from the identity representations of two‐block Young subgroups are Markov and find explicit formulas for the transition probabilities of the corresponding Markov measures. The induced representations are studied with the help of the tensor model of two‐row representations of the symmetric groups; in particular, we deduce explicit formulas for the Gelfand–Tsetlin basis in the tensor models.

A formula of Lascoux-Leclerc-Thibon and representations of symmetric groups

We consider Green polynomials at roots of unity, corresponding to partitions which we call l-partitions. We obtain a combinatorial formula for Green polynomials corresponding to l-partitions at primitive lth roots of unity. The formula is rephrased in terms of representation theory of the symmetric group.

Completely splittable representations of symmetric groups and affine Hecke algebras

We study the class of completely splittable representations of the symmetric group and its associated degenerate affine Hecke algebra in positive characteristic, using techniques developed by Okounkov and Vershik for the symmetric group over the complex numbers. The representations are classified and constructed, and we provide a character formula in the symmetric group case.

Weight two blocks of Iwahori–Hecke algebras of type B

In the representation theory of Iwahori–Hecke algebras of type A (and in particular for representations of symmetric groups) the notion of the weight of a block, introduced by James, plays a central rôle. Richards determined the decomposition numbers for blocks of weight 2, and here the same task is undertaken for weight two blocks of Iwahori–Hecke algebras of type B, using the author's own definition of the weight of a bipartition.

On different models of representations of the infinite symmetric group

We present an explicit description of the isomorphism between two models of finite factor representations of the infinite symmetric group: the tableau model in the space of functions on Young bitableaux and the dynamical model in the space of functions on pairs of Bernoulli sequences. The main tool used is the Fourier transform on the symmetric groups. We also start the investigation of the so-called tensor model of two-row representations of the symmetric groups, which plays an intermediate role between the tableau and dynamical models, and show its relations to both these models.

Character formulas for the operad of two compatible brackets and for the bi-Hamiltonian operad

We compute the dimensions of the components for the operad of two compatible brackets and for the bi-Hamiltonian operad. We also obtain character formulas for the representations of symmetric groups and SL 2 in these spaces.

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.

STRATIFYING SYSTEMS, FILTRATION MULTIPLICITIES AND SYMMETRIC GROUPS

We show that the theorem by Hemmer and Nakano, on uniqueness of Specht filtration multiplicities, can be proved working entirely with representations of symmetric groups, or Hecke algebras. Furthermore, we give a new proof that Schur algebras are quasi-hereditary provided the characteristic of the field is at least 5. Our tools are some more general results on stratifying systems.

On relations between the classical and the Kazhdan–Lusztig representations of symmetric groups and associated Hecke algebras

Let H be the Hecke algebra of a Coxeter system ( W , S ) , where W is a Weyl group of type A n , over the ring of scalars A = Z [ q 1 / 2 , q - 1 / 2 ] , where q is an indeterminate. We show that the Specht module S λ , as defined by Dipper and James [Proc. London Math. Soc. 52(3) (1986) 20–52], is naturally isomorphic over A to the cell module of Kazhdan and Lusztig [Invent. Math. 53 (1979) 165–184] associated with the cell containing the longest element of a parabolic subgroup W J for appropriate J ⊆ S . We give the association between J and λ explicitly. We introduce notions of the T-basis and C-basis of the Specht module and show that these bases are related by an invertible triangular matrix over A. We point out the connection with the work of Garsia and McLarnan [Adv. Math. 69 (1988) 32–92] concerning the corresponding representations of the symmetric group.

The Hyperoctahedral Group, Symmetric Group Representations and the Moments of the Real Wishart Distribution

In this paper, we compute all the moments of the real Wishart distribution. To do so, we use the Gelfand pair (S2k,H), where H is the hyperoctahedral group, the representation theory of H and some techniques based on graphs.

Atomic spectral methods for molecular electronic structure calculations

Theoretical methods are reported for ab initio calculations of the adiabatic (Born-Oppenheimer) electronic wave functions and potential energy surfaces of molecules and other atomic aggregates. An outer product of complete sets of atomic eigenstates familiar from perturbation-theoretical treatments of long-range interactions is employed as a representational basis without prior enforcement of aggregate wave function antisymmetry. The nature and attributes of this atomic spectral-product basis are indicated, completeness proofs for representation of antisymmetric states provided, convergence of Schrodinger eigenstates in the basis established, and strategies for computational implemention of the theory described. A diabaticlike Hamiltonian matrix representative is obtained, which is additive in atomic-energy and pairwise-atomic interaction-energy matrices, providing a basis for molecular calculations in terms of the (Coulombic) interactions of the atomic constituents. The spectral-product basis is shown to contain the totally antisymmetric irreducible representation of the symmetric group of aggregate electron coordinate permutations once and only once, but to also span other (non-Pauli) symmetric group representations known to contain unphysical discrete states and associated continua in which the physically significant Schrodinger eigenstates are generally embedded. These unphysical representations are avoided by isolating the physical block of the Hamiltonian matrix with a unitary transformation obtained from the metric matrix of the explicitly antisymmetrized spectral-product basis. A formal proof of convergence is given in the limit of spectral closure to wave functions and energy surfaces obtained employing conventional prior antisymmetrization, but determined without repeated calculations of Hamiltonian matrix elements as integrals over explicitly antisymmetric aggregate basis states. Computational implementations of the theory employ efficient recursive methods which avoid explicit construction the metric matrix and do not require storage of the full Hamiltonian matrix to isolate the antisymmetric subspace of the spectral-product representation. Calculations of the lowest-lying singlet and triplet electronic states of the covalent electron pair bond (H(2)) illustrate the various theorems devised and demonstrate the degree of convergence achieved to values obtained employing conventional prior antisymmetrization. Concluding remarks place the atomic spectral-product development in the context of currently employed approaches for ab initio construction of adiabatic electronic eigenfunctions and potential energy surfaces, provide comparisons with earlier related approaches, and indicate prospects for more general applications of the method.

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.