Double cosets in F4 Lawther, R.
Journal of algebra,
10/2022, Letnik:
607
Journal Article
Recenzirano
In this paper we let G be the finite simple group F4(q), for q a prime power; we take P to be the maximal parabolic subgroup with Levi subgroup having derived group B3(q), and H to be the subgroup ...D4(q) generated by the long root elements of G. We consider the collection of (P,H)-double cosets in G; we obtain coset representatives and give the decomposition explicitly. As a consequence we are able to deduce that if q>2 then the action of F4(q) on the cosets of the maximal subgroup D4(q).S3 is not multiplicity-free, even if field automorphisms are applied.
Let H and K be subgroups of a finite group G. Pick g∈G uniformly at random. We study the distribution induced on double cosets. Three examples are treated in detail: 1) H=K= the Borel subgroup in ...GLn(Fq). This leads to new theorems for Mallows measure on permutations and new insights into the LU matrix factorization. 2) The double cosets of the hyperoctahedral group inside S2n, which leads to new applications of the Ewens's sampling formula of mathematical genetics. 3) Finally, if H and K are parabolic subgroups of Sn, the double cosets are ‘contingency tables’, studied by statisticians for the past 100 years.
An algorithm is developed for calculating the number of double cosets P﹨Sn/P, where P is a Sylow-p-subgroup of the symmetric group Sn. Several examples of its use are given.
Extending Camina pairs Akhlaghi, Zeinab; Beltrán, Antonio
Journal of algebra,
05/2023, Letnik:
622
Journal Article
Recenzirano
Odprti dostop
Let G be a finite group and N a nontrivial proper normal subgroup of G. A.R. Camina introduced the class of finite groups G, which extends Frobenius groups, satisfying that for all g∈G−N and n∈N, gn ...is conjugate to g. He proved that under these assumptions one of three possibilities occurs: G is a Frobenius group with kernel N; or N is a p-group; or G/N is a p-group. In this paper we extend this class of groups by investigating the structure of those finite groups G having a nontrivial proper normal subgroup N such that gn is conjugate to either g or g−1 for all g∈G−N and all n∈N.
In Journal of Pure and Applied Algebra 224 (2020), no 12, 106449, V. Mazorchuk and R. Mrđen (with some help by A. Hultman) prove that, given a Weyl group, the intersection of a Bruhat interval with a ...parabolic coset has a unique maximal element and a unique minimal element. We show that such intersections are actually Bruhat intervals also in the case of an arbitrary Coxeter group.
One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. Quantum maximal distance separable (MDS) codes are optimal in the sense they attain ...maximal minimum distance. Recently, constructing quantum MDS codes has received much attention and seems to become more and more difficult. In this paper, based on classical constacyclic codes, we construct some new quantum MDS codes by employing the Hermitian construction. Compared with the known quantum MDS codes, these quantum MDS codes have much larger minimum distance.
Let F be a finite field. Consider a direct sum V of an infinite number of copies of F, consider the dual space V⋄, i.e., the direct product of an infinite number of copies of F. Consider the direct ...sum V=V⋄⊕V. The object of the paper is the group GL‾ of continuous linear operators in V. We reduce the theory of unitary representations of GL‾ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over F. In fact we consider a certain family Q‾α of subgroups in GL‾ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to Q‾α, and reduce this multiplication to products of linear relations. We show that this group has type I and obtain an ‘upper estimate’ of the set of all irreducible unitary representations of GL‾.
The last decades, mark an accelerated progression in the determination of the parameters of the primitive BCH codes. Indeed, BCH codes are powerful in terms of decoding. They are applied in several ...fields such as: satellite communications, cryptography, compact disk drives... and have good structural properties. Nevertheless, the dimension and the minimum distance of those codes aren't known, in general. In this paper, we present a class of narrow sense primitive BCH codes of designed distance $\delta_{_4}=(q-1)q^{^{m-1}}-1-q^{\lfloor \frac{m+3}{2 }\rfloor}.$ Also, we investigate their Bose distance and the dimension.
The entanglement-assisted (EA) formalism allows arbitrary classical linear codes to transform into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement ...between the sender and the receiver. In this work, we propose a decomposition of the defining set of constacyclic codes. Using this method, we construct four classes of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of q-ary EAQMDS have minimum distance upper bound greater than 3q−1. Some of them have much larger minimum distance than the known quantum MDS (QMDS) codes of the same length. Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.
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