A note on Cartan matrices for symmetric groups

Using generating functions, a very simple explicit formula for the determinants of the p-Cartan matrices of symmetric groups is given. Our method works also when p is a composite number.Received: 5 September 2001     

Block characters of the symmetric groups

A block character of a finite symmetric group is a positive definite function which depends only on the number of cycles in a permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the characters to descent representations and the coinvariant algebra of ${\mathfrak{S}}_{n}$ . The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group ${\mathfrak{S}}_{\infty}$ , along with their connection to the Thoma characters of the infinite linear group GL _∞(q) over a Galois field.     

Spin characters of generalized symmetric groups

In 1911 Schur computed the spin character values of the symmetric group using two important ingredients: the first one later became famously known as the Schur Q-functions and the second one was certain creative construction of the projective characters on Clifford algebras. In the context of the McKay correspondence and affine Lie algebras, the first part was generalized to all wreath products by the vertex operator calculus in Frenkel et al. (Duke Math J 111:51–96, 2002) where a large part of the character table was produced. The current paper generalizes the second part and provides the missing projective character values for the wreath product of the symmetric group with a finite abelian group. Our approach relies on Mackey–Wigner’s little groups to construct irreducible modules. In particular, projective modules and spin character values of all classical Weyl groups are obtained.     

Arithmetic of characters of generalized symmetric groups

The result here answers the following questions in the affirmative: Can the Galois action on all abelian (Galois) fields $K/\mathbb{Q}$ be realized explicitly via an action on characters of some finite group? Are all subfields of a cyclotomic field of the form $\mathbb{Q}(\chi)$, for some irreducible character $\chi$ of a finite group G? In particular, we explicitly determine the Galois action on all irreducible characters of the generalized symmetric groups. We also determine the smallest extension of $\mathbb{Q}$ required to realize (using matrices) a given irreducible representation of a generalized symmetric group. Received: 18 February 2002     

Upper bound on the characters of the symmetric groups

Let C be a conjugacy class in the symmetric group S _n , and λ be a partition of n. Let f ^λ be the degree of the irreducible representation S ^λ , χ ^λ (C)– the character of S ^λ at C, and r ^λ (C)– the normalized character χ ^λ (C) f ^λ . We prove that there exist constants b > 0 and 1 > q > 0 such that for n > 4, for every conjugacy class C in S _n and every irreducible representation S ^λ of S _n ∣r ^λ (C)∣≦ ( max {q,λ ₁ n, λ ₁ ′ n}) ^{b ⋅ supp(C)} where supp (C) is the number of non-fixed digits under the action of a permutation in C, λ ₁ is the size of the largest part in λ, and λ ₁ ′ is the number of parts in λ. The proof is obtained by enumeration of rim hook tableaux, the Hook formula and probabilistic arguments. Combinatorial, algebraic and statistical applications follow this result. In particular, we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes. Oblatum 14-III-1995 & 30-X-1995     

A note on generalized characters

For a compactly generated LCA group G, it is shown that the setH(G) of all generalized characters on G equipped with the compact-open topology is a LCA group andH(G) = Ĝ (the dual group ofG) if and only ifG is compact. Both results fail for arbitrary LCA groups. Further, ifG is second countable, then the Gel’fand space of the commutative convolution algebraC _c (G) equipped with the inductive limit topology is topologically homeomorphic toH(G).     

A note on insurance leverage under skew distributions

Insurance leverage denotes the impact of the insurance specific reserves on the return on equity of the insurance company. We analyze insurance leverage on the basis of a stochastic one-period model of the insurance business, especially accounting for the skewness of the involved distributions.     

A note on labeled symmetric graphs

By using the known results of groups and graphs, we determine the number of labeled symmetric graphs with a prime number of vertices.     

A note on symmetric doubly-stochastic matrices

Each extreme point in the convex set Δ¹_n of all n×n symmetric doubly-stochastic matrices is shown to have the form $\text{1}\text{2}\text{(P + P}{}^{\mathrm{t}}\text{)}$, for some n×n permutation matrix P. The convex hull Σ_n of the integral points in Δ¹_n (i.e., the symmetric permutation matrices) is shown to consist of those matrices, X = (x_ij) in Δ¹_n satisfying Σ_i∈SΣ_j∈S?{i}x_ij ? 2k, for each subset S of {1, 2, …, n} having cardinality 2k + 1, for some k > 0.     

A note on generalized skew derivations on Lie ideals

Let \(\mathcal {R}\) be a prime ring, \(\mathcal {Z(R)}\) its center, \(\mathcal {C}\) its extended centroid, \(\mathcal {L}\) a Lie ideal of \(\mathcal {R}, \mathcal {F}\) a generalized skew derivation associated with a skew derivation d and automorphism \(\alpha \). Assume that there exist \(t\ge 1\) and \(m,n\ge 0\) fixed integers such that \( vu = u^m\mathcal {F}(uv)^tu^n\) for all \(u,v \in \mathcal {L}\). Then it is shown that either \(\mathcal {L}\) is central or \(\mathrm{char}(\mathcal {R})=2, \mathcal {R}\subseteq \mathcal {M}_2(\mathcal {C})\), the ring of \(2\times 2\) matrices over \(\mathcal {C}, \mathcal {L}\) is commutative and \(u^2\in \mathcal {Z(R)}\), for all \(u\in \mathcal {L}\). In particular, if \(\mathcal {L}=[\mathcal {R,R}]\), then \(\mathcal {R}\) is commutative.     

A note on skew differential operators on commutative rings

Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module over the fixed sub-ring A ^ø, with a basis containing 1, then D(A;ø) is isomorphic to the matrix ring M_m(D(A ^ø). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c?Asuch that D(A[c^-1];ø)?M _m(D(A[c^-1]^ø)). As an application, we consider the structure of D(A;ø)when A is a polynomial or Laurent polynomial ring over k and ø is a diagonalizable linear automorphism.     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

     

Gaussian fluctuations of characters of symmetric groups and of Young diagrams

We study asymptotics of reducible representations of the symmetric groups S _q for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly chosen Young diagram). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian; in this way we generalize Kerov's central limit theorem. The considered class consists of representations for which the characters almost factorize and this class includes, for example, the left-regular representation (Plancherel measure), irreducible representations and tensor representations. This class is also closed under induction, restriction, outer product and tensor product of representations. Our main tool in the proof is the method of genus expansion, well known from the random matrix theory.     

On strongly symmetric skew lattices

Skew lattices are a non-commutative generalization of lattices. In the past 20?years, several varieties of skew lattices have been introduced. In the present paper we study the variety of strongly symmetric skew lattices.     

A note on multiobjective second-order symmetric duality

In this paper, we establish a strong duality theorem for a pair of multiobjective second-order symmetric dual programs. This removes an omission in an earlier result by Yang et al. [X.M. Yang, X.Q. Yang, K.L. Teo, S.H. Hou, Multiobjective second-order symmetric duality with F-convexity, Euro. J. Oper. Res. 165 (2005) 585–591].     

A note on homogeneous locally symmetric spaces

Let denote an orthogonal symmetric Lie algbra and let (G, K) be an associated pair, i.e., Lie(G = and Lie(K°) = . In this paper we prove that the homogeneous spaceG/K has a structure of a globally symmetric space for every choice ofG andK, especially forG being compact.