Revisiting the Marcus–de Oliveira Conjecture

Let p p be a prime. We denote by S n S_n the symmetric group of degree n n , by A n A_n the alternating group of degree n n and by F p {\mathbb F}_p the field with p p elements. An important concept of modular representation theory of a finite group G G is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring F q G {\mathbb F}_q G , where q q is a prime power. Here, we describe a new method to compute the primitive central idempotents of F q G {\mathbb F}_q G for arbitrary prime powers q q and arbitrary finite groups G G . For the group rings F p S n {\mathbb F}_p S_n of the symmetric group, we show how to derive the primitive central idempotents of F p S n − p {\mathbb F}_p S_{n-p} from the idempotents of F p S n {\mathbb F}_p S_n . Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of F p S n {\mathbb F}_p S_n which contains the primitive central idempotents. The described results are most efficient for p = 2 p = 2 . In an appendix we display all primitive central idempotents of F 2 S n {\mathbb F}_2 S_n and F 4 A n {\mathbb F}_4 A_n for n ≤ 50 n \le 50 which we computed by this method.

The symmetric group given by a Gröbner basis

Let be the cyclic group of order m and N be the direct product of n copies of . Let Sn be the symmetric group of degree n. The wreath product of with Sn is a split extension of N by Sn, called the generalized symmetric group, here denoted by B(m, n). In his Ph.D. thesis Almestady presented a combinatorial method for constructing the Fischer-Clifford matrices of B(m, n). However as a few examples for small values of m and n show, the manual calculation of these matrices presents formidable problems and hence a computerized approach to this combinatorial method is necessary. In a previous paper the current authors have given a computer programme that computes matrices which are row equivalent to the Fischer-Clifford matrices of B(2, n). Here that programme is generalized to B(m, n), where m is any positive integer. It is anticipated that with some improvements, a number of the programmes given here can be incorporated into GAP. Indeed with further development work these programmes should lead to an alternative method for computing the character table of B(m, n) in GAP.

We construct two kinds of presentations for the alternating and symmetric groups of degree n n : the first are on two generators in which the number of relations is O ( log ⁡ n ) O(\log n) and the presentation length is O ( log 2 ⁡ n ) O(\log ^2 n) ; the second have a bounded number of generators and relations and length O ( log ⁡ n ) O(\log n) .

Symmetric 1-factorizations of the complete graph

Extraspecial towers and weil representations

The uniqueness of a certain graph

Let\n$G$ be a subgroup of\n$S_{6}$ , the symmetric group of degree 6. For any field\n$k$ ,\n$G$ acts naturally on the rational function field\n$k(x_{1},\\ldots,x_{6})$ via\n$k$ -automorphisms defined by\n$\\sigma\\cdot x_{i}=x_{\\sigma(i)}$ for any\n$\\sigma\\in G$ and any\n$1\\le i\\le 6$ . We prove the following theorem. The fixed field\n$k(x_{1},\\ldots,x_{6})^{G}$ is rational (i.e., purely transcendental) over\n$k$ , except possibly when\n$G$ is isomorphic to\n$\\operatorname{PSL}_{2}(\\mathbb{F}_{5})$ ,\n$\\operatorname{PGL}_{2}(\\mathbb{F}_{5})$ , or\n$A_{6}$ . When\n$G$ is isomorphic to\n$\\operatorname{PSL}_{2}(\\mathbb{F}_{5})$ or\n$\\operatorname{PGL}_{2}(\\mathbb{F}_{5})$ , then\n$\\mathbb{C}(x_{1},\\ldots,x_{6})^{G}$ is\n$\\mathbb{C}$ -rational and\n$k(x_{1},\\ldots,x_{6})^{G}$ is stably\n$k$ -rational for any field\n$k$ . The invariant theory of wreath products will be investigated also.

Rational invariants for subgroups of [formula omitted] and [formula omitted

The modules and are described for certain simple -modules (the completely splittable ones or close to them), where is a field of characteristic and is the symmetric group of degree . This result is based on an upper bound for the dimensions of the -spaces between certain simple modules.

Let Gn,t be the subgroup of GL(n,ℤ2) that stabilizes {xℤ2n:|x|≤t}. We determine Gn,t explicitly: For 1≤t≤n−2, Gn,t=Sn when t is odd and Gn,t=〈Sn,Δ〉 when t is even, where Sn<GL(n,ℤ2) is the symmetric group of degree n and ΔGL(n,ℤ2) is a particular involution. Let ℛn,t be the set of all binary t-resilient functions defined on ℤ2n. We show that the subgroup ℤ2n⋊(Gn,t∪Gn,n−1−t)<AGL(n,ℤ2) acts on ℛn,t/ℤ2. We determine the representatives and sizes of the conjugacy classes of ℤ2n⋊Sn and ℤ2n⋊〈Sn,Δ〉. These results allow us to compute the number of orbits of ℛn,t/ℤ2 under the above group action for (n,t)=(5,1) and (6,2).

Permutations with pl-th roots

Some identities of symmetry for generalized Carlitz-type q-Euler polynomials under the symmetric group of degree four

With the assumption that baryon resonance mass differences are mainly attributed to kinetic energy differences and partly to spin orbit coupling perturbations, the baryon resonance mass level pattern for unitary octets, decuplets, and singlets with various spin and parity values is estimated by classification with symmetric group of degree three and compared with experimental data. Reasonable agreement indicates the assumption may be reasonable. It is assumed that the over-all wave function is totally symmetric and the internal (vibrational) wave function is also totally symmetric for permutation.

Inequalities for the q-permanent. II