Irreducible Characters Of Sn Research Articles

The second immanantal polynomials of Laplacian matrices of unicyclic graphs

Let M=(aij) be a square matrix of order n. The second immanant of M isd2(M)=∑σ∈Snχ(σ)∏t=1natσ(t),where χ is the irreducible character of Sn corresponding to the partition (2,1n−2). Let G be a graph. Denote by L(G) the laplacian matrix of G. The polynomial ϕ(G,x)=d2(xI−L(G))=∑k=0n(−1)kck(G)xn−k is called the second immanantal polynomial of G, where ck(G) is the k-th coeficient of ϕ(G,x). The coefficient ck(G) admit various algebraic and topological interpretations for a graph. Merris first considered the coefficient cn−1(G), and he proposed an open problem: given n and m, can the graph G be determined for which cn−1(G) is a minimum or maximum? In this article, we investigate the problem, and we construct six graph operations each of which changes the value of cn−1(G). Using these graph operations, we determine the bound of cn−1(G) of unicyclic graphs. In particular, we also determined the corresponding extremal graphs.

Kronecker positivity and 2-modular representation theory

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl’s conjecture for several large new families of partitions. In particular, we verify Saxl’s conjecture for all irreducible characters of S n \mathfrak {S}_n which are of 2-height zero.

The probability that a character value is zero for the symmetric group

Let be chosen at random from the irreducible characters of the symmetric group Sn and let g be chosen at random from the group itself. What is the probability that .g/ D 0? In this short note we give a remarkable asymptotic answer of one. Throughout the paper “at random” means uniformly at random. Theorem 1. If is chosen at random from the irreducible characters of Sn and g is chosen at random from Sn, then .g/ D 0 with probability P.Sn/ ! 1 as n ! 1. It will follow that the same must be true for the alternating group An. Theorem 2. If is chosen at random from the irreducible characters ofAn and g is chosen at random from An, then .g/ D 0 with probability P.An/ ! 1 as n ! 1. We prove these results in Section 1 and make some remarks in Section 2.

ON CONSTRUCTING REPRESENTATIONS OF THE SYMMETRIC GROUPS

Let G be a symmetric group. In this paper we describe a method that for a certain irreducible characterof G it flnds a subgroup H such that the restriction ofon H has a linear con- stituent with multiplicity one. Then using a well known algorithm we can construct a representation of G afiording ´. m ), l1 > ¢¢¢ > lm > 0; when we have ai parts of size li. Since the number of irreducible characters of a group is equal to the number of conjugacy classes, which in the case of Sn is the number of partitions of n, the irreducible characters of Sn are labelled by partitions of n. We denote the irreducible character labelled by the partition ‚ by (‚), so Irr(Sn) = f(‚) j ‚ ' ng. If G is a flnite group andis an irreducible character of G, an e-cient and simple method to construct representations of flnite groups has been presented in (2). This method is applicable whenever G has a subgroup H such thatH has a linear constituent with multiplicity 1. We call such a subgroup H a ´-subgroup. In practice this algorithm is quite fast when H has a small order, but can be very slow for a large H. For using this method to construct representations of G, we need to flnd a ´-subgroup for each irreducible characterof G. If G is a simple group or a covering group of a simple group, then a ´- subgroup for each nontrivial irreducible characterof G of degree < 32 has been found in (1). Also if ‚ is a partition of n and ‚ 0 is the conjugate

Zeros in tables of characters for the groups Sn and An

In the representation theory of symmetric groups, for each partition α of a natural number n, the partition h(α) of n is defined so as to obtain a certain set of zeros in the table of characters for Sn. Namely, h(α) is the greatest (under the lexicographic ordering ≤) partition among β ∈ P(n) such that χα(gβ) ≠ 0. Here, χα is an irreducible character of Sn, indexed by a partition α, and gβ is a conjugacy class of elements in Sn, indexed by a partition β. We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition α ∈ P(n), the partition f(α) of n is defined so that f(α) is greatest among the partitions β of n which are opposite in sign to h(α) and are such that χα(gβ) ≠ 0 (Thm. 1). Also, for any self-associated partition α of n > 1, we construct a partition $$\tilde f$$ (α) ∈ P(n) such that $$\tilde f$$ (α) is greatest among the partitions β of n which are distinct from h(α) and are such that χα(gβ) ≠ 0 (Thm. 2).

On the Irreducible Characters of the Groups Snand An

Two characters ϕ and ψ of a finite group G are called semiproportional if they are not proportional and there exists a set M in G such that the restrictions of ϕ and ψ to M and G∖M are proportional. We obtain a description for all pairs of proportional irreducible characters of symmetric groups. Namely, in Theorem 1 we prove equivalence of the following conditions for a pair (ϕ,ψ) of different irreducible characters of Sn (n∈ N): (1) ϕ and ψ are semiproportional; (2) ϕ and ψ have the same roots; and (3) ϕ are ψ are associated (i.e., ψ=ϕξ where ξ is a linear character of Sn with kernel An). Note that (1) and (2) are in general not equivalent for arbitrary finite groups. For the symmetric groups, the equivalence of (1) and (3) validates the following conjecture proven earlier by the author for a number of group classes: semiproportional irreducible characters of a finite group have the same degree. The alternating groups seem to have no semiproportional irreducible characters. Theorem 2 of this article is a step towards proving this conjecture.

LITTLEWOOD'S MULTIPLE FORMULA FOR SPIN CHARACTERS OF SYMMETRIC GROUPS

This paper deals with some character values of the symmetric group Sn as well as its double cover ~Sn. Let xλ(p) be the irreducible character of Sn, indexed by the partition λ and evaluated at the conjugacy class p. Comparing the character tables of S2 and S4, one observes that x(4)(2p)=x(2)(p) x(22)(2p)=x(2)(p)+x(1(2))(p) for p = (2), 2p = (4) and p = (12), 2p = (22). A number of such observations lead to what we call Littlewood's multiple formula (Theorem 1.1). This formula appears in Littlewood's book [2]. We include a proof that is based on an `inflation' of the variables in a Schur function. This is different from one given in [2], and we claim that it is more complete than the one given there. Our main objective is to obtain the spin character version of Littlewood's multiple formula (Theorem 2.3). Let ζλ(p) be the irreducible negative character of ~Sn (cf. [1]), indexed by the strict partition λ and evaluated at the conjugacy class p. One finds character tables (ζλ(p)) in [1] for n ≤ 14. This time we evidently see that ζ3λ(3p) = ζλ(p) for λ = (4),(3; 1) and p = (3,1),(14). The proof of Theorem 2.3 is achieved in a way that is similar to the case of ordinary characters. Instead of a Schur function, we deal with Schur's P-function, which is defined as a ratio of Pfaffians.

Spherical functions and immanants

Let λ be an irreducible character of Sn corresponding to the partition (r,s) of n. Let A be a positive semidefinite Hermitian n × n matrix. Let dλ (A) and per(A) be the immanants corresponding to λ and to the trivial character of Sn , respectively. A proof of the inequality dλ(A)≤λ(id)per(A) is given.

A note on singular matrices satisfying certain polynomial identities

Let Sn be the symmectric group of degree n G a subgroup of Sn a field and λ and -valued character of G. We describe the singular matrices A that satisfy and if χ is an irreducible character of Sn , we describe the singular matrices A and B such that

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If a = {ava2, ..., 0, is a partition of n then Xa denotes the associated irreducible character of Sn, the symmetric group on {1,2,...,«}, and, if ceCSn, the group algebra generated by C and Sn, then dc() denotes the generalized matrix function associated with c. If cvc2eCSn then we write Cj = 1 and as = 2, and a.' denotes {a^otj, . . . ,a8_j, 1'} then ^a ^ (K ® {2))* where f denotes character induction from Sn_2 x S2 to 5n . This in turn implies that if a = {«1,a2,...,as,l } with s> 1, as = 2, and /? denotes {a1 + 2,a2, ...,as_,, 1'} then Xa =< ip which, in conjunction with other known results, provides many new inequalities among immanants. In particular it implies that the permanent function dominates all normalized immanants whose associated partitions are of rank 2, a result which has proved elusive for some years. We also consider the non-relationship problem for immanants that is the problem of identifying pairs, (a,ff) such that Xa = A« and kg ^ Aa are both false.

A generalization of hadamard's inequality

If A = (ai,j ) is an n × n complex matrix then h(A) denotes the product of the diagonal entries of A, and if λ is a partition of n then [λ](A) is defined by where Sn denotes the symmetric group of degree n and {λ} is the ordinary irreducible character of Sn corresponding to λ. Let deg λ denote the degree of {λ}. In this paper we investigate which partitions λ of n satisfy the inequality for all n × n Hermitian positive semi-definite matrices A. When λ = (1 n ) the inequality was proved by J. Hadamard. In this paper we show, amongst other things, that the inequality holds for: .