Narrow Results

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

This paper studies mathematical properties of reaction systems that were introduced by Ehrenfeucht and Rozenberg as computational models inspired by biochemical reaction in the living cells. In particular, we continue the study on the generative power of functions specified by minimal reaction systems under composition initiated by Salomaa. Allowing degenerate reaction systems, functions specified by minimal reaction systems over a quarternary alphabet that are permutations generate the alternating group on the power set of the background set.

Let $n\in \mathbb{Z}$ with $n\ge 3$. Let $S_n$ and $A_n$ denote, respectively, the symmetric group and alternating group on $n$ letters. Let $m$ be an indeterminate, and define

Let ${{𝒮}}_N$ be symmetric group of degree $N$. In the representation of ${{𝒮}}_N$ corresponding to trapeziform partition (precisely $\tau =(m^{2k},a,b)$, $0\le b\le a<m$), we propose nonsymmetric Jack polynomials in $2mk+a+b$ variables which are singular at $\frac{1}{m+2}$. Moreover those nonsymmetric Jack polynomials span a module of ${{𝒮}}_N$ which is isomorphic to the representation of type $\sigma =(mk+a,mk+b)$.

We describe how Computational Group Theory provides tools for manipulating indexed objects (tensors, spinors, and so on) in explicit index notation. In particular, we present algorithms that put tensors with free and dummy indices obeying permutation symmetries into the canonical form. The method is based on algorithms for determining canonical double coset representatives of subgroups of the symmetric group. The complexity of the algorithms is polynomial on the number of indices in practical applications and allows one to address the simplification of tensor expressions with hundreds of indices, going beyond what is needed in practical applications.

The conjugacy classes of any group are important since they reflect some aspects of the structure of the group. The construction of the conjugacy classes of finite groups has been a subject of research for several authors. Let n,m be positive integers and be the direct product of m copies of the symmetric group S_n of degree n. Then is a subgroup of the symmetric group S_mn of degree m × n. Let g∈S_mn, of type [mⁿ] where each m-cycle contains one symbol from each set of symbols in that order on which the copies of S_n act. Then g permutes the elements of the copies of S_n in and generates a cyclic group C_m = 〈g〉 of order m. The wreath product of S_n with C_m is a split extension or semi-direct product of by C_m, denoted by . It is clear that is a subgroup of the symmetric group S_mn. In this paper we give a method similar to coset analysis for constructing the conjugacy classes of , where m is prime. Apart from the fact that this is an alternative method for constructing the conjugacy classes of the group , this method is useful in the construction of Fischer–Clifford matrices of the group . These Fischer–Clifford matrices are useful in the construction of the character table of .

By using simple ideas from subgroup growth of pro-finite groups we deduce some combinatorial identities on generating functions counting various elements in symmetric groups.

The W-graph representations for symmetric groups G_n and their Hecke algebras have been studied by many authors, but the situation for a large n (for instance n ≥ 16) has not been completely solved. The aim of this note is to provide an algorithm of "inducing W-graphs" step by step from smaller n, to construct W-graphs for the left cells of the symmetric groups for bigger n.

We give a polynomial basis of each irreducible representation of the Hecke algebra, as well as an adjoint basis. Decompositions in these bases are obtained by mere specializations.

We show that, like in the case of algebras over fields, the study of multilinear polynomial identities of unitary rings can be reduced to the study of proper polynomial identities. In particular, the factors of series of ℤS_n-submodules in the ℤS_n-modules of multilinear polynomial functions can be derived by the analog of Young's (or Pieri's) rule from the factors of series in the corresponding ℤS_n-modules of proper polynomial functions. As an application, we calculate the codimensions and a basis of multilinear polynomial identities of unitary rings of upper triangular 2 × 2 matrices and infinitely generated Grassmann algebras over unitary rings. In addition, we calculate the factors of series of ℤS_n-submodules for these algebras. Also we establish relations between codimensions of rings and codimensions of algebras and show that the analog of Amitsur's conjecture holds in all torsion-free rings, and all torsion-free rings with 1 satisfy the analog of Regev's conjecture.

This paper treats algebras whose support $\tau$-tilting posets are isomorphic to the poset of the symmetric group with the weak order. We determine such algebras in terms of quivers and relations.

In this paper, we present the results of a computer investigation of asymptotics for maximum dimensions of linear and projective representations of the symmetric group. This problem reduces the investigation of standard and strict Young diagrams of maximum dimensions. We constructed some sequences for both standard and strict Young diagrams with extremely large dimensions. The conjecture that the limit of normalized dimensions exists was proposed by Vershik 30 years ago [A. M. Vershik and S. V. Kerov, Funktsional Anal. i Prilozhen19(1) (1985) 25–36] and has not been proved yet. We studied the growth and oscillations of the normalized dimension function in sequences of Young diagrams. Our approach is based on analyzing finite differences of their normalized dimensions. This analysis also allows us to give much more precise estimation of the limit constants.

The braids of $B_{\infty }$ can be equipped with a self-distributive operation ⊳ enjoying a number of deep properties. This text is a survey of known properties and open questions involving this structure, its quotients, and its extensions.

Gaussian and Poisson noises can be discussed not only in parallel, but we can also see dissimilarity between them together with some interesting results by using the space of quadratic Hida distributions. Further we can discuss calculus of their functionals to see a duality of the two noises.

Using the structure of the Boson-Fermion Fock space and an argument taken from [P. Bieliavsky, M. Cahen, S. Gutt, J. Rawnsley and L. Schwachhofer, Symplectic connections, Int. J. Geom. Meth. Mod. Phys.3 (2006) 375–420], we give a new proof of the triviality of the L² cohomology groups on an abstract Wiener space, alternative to that given by Shigekawa [De Rham–Hodge–Kodaira's decomposition on an abstract Wiener space, J. Math. Kyoto. Univ.26(2) (1986) 191–202]. We apply the representation theory of the symmetric group to characterize the spaces of exact and co-exact forms in their Boson-Fermion Fock space representation.

This paper surveys, and in some cases generalizes, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined.

The commuting graph of a group G, denoted by Γ(G), is a simple undirected graph whose vertices are all non-central elements of G and two distinct vertices x, y are adjacent if xy = yx. The commuting graph of a subset of a group is defined similarly. In this paper we investigate the properties of the commuting graph of the symmetric and alternating and subsets of transpositions and involutions in the symmetric groups.

Let ℤG be the integral group ring of the finite nonabelian group G over the ring of integers ℤ, and let * be an involution of ℤG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (u_{k, m}(x), u_{k, m}(x*)) or (u_{k, m}(x), u_{k, m}(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ℤG.

Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair if h = [h,_n g] and g = [g,_n h]. In this paper, we will study the subgroup generated by the n-Engel pair under certain conditions.

Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair, n ≥ 2, if h = [h,_n g], g = [g,_n h] and h ≠ 1. In this paper, we prove that if (h, g) is an n-Engel pair, hgh^-2gh = ghg and ghg^-2hg = hgh, then n = 2k where k = 4 or k ≥ 6. Furthermore, the subgroup generated by {h, g} is determined for k = 4, 6, 7 and 8.