Search results for "symmetric group"
showing 10 items of 43 documents
Some results concerning simple locally finite groups of 1-type
2005
AbstractIn this paper several aspects of infinite simple locally finite groups of 1-type are considered. In the first part, the classes of diagonal limits of finite alternating groups, of diagonal limits of finite direct products of alternating groups, and of absolutely simple groups of 1-type are distinguished from each other. In the second part, inductive systems of representations over fields of characteristic zero (which are known to correspond to ideals in the group algebra) are studied in general for groups of 1-type. The roles of primitive respectively imprimitive representations in inductive systems are investigated. Moreover it is shown that in any proper inductive system the depth…
Some problems in number theory that arise from group theory
2021
In this expository paper, we present several open problems in number theory that have arisen while doing research in group theory. These problems are on arithmetical functions or partitions. Solving some of these problems would allow to solve some open problem in group theory.
Character restrictions and multiplicities in symmetric groups
2017
Abstract We give natural correspondences of odd-degree characters of the symmetric groups and some of their subgroups, which can be described easily by restriction of characters, degrees and multiplicities.
Computing the ℤ2-Cocharacter of 3 × 3 Matrices of Odd Degree
2013
Let F be a field of characteristic 0 and A = M 2, 1(F) the algebra of 3 × 3 matrices over F endowed with the only non trivial ℤ2-grading. Aver'yanov in [1] determined a set of generators for the T 2-ideal of graded identities of A. Here we study the identities in variables of homogeneous degree 1 via the representation theory of the symmetric group, and we determine the decomposition of the corresponding character into irreducibles.
A characterization of fundamental algebras through S-characters
2020
Abstract Fundamental algebras play an important role in the theory of algebras with polynomial identities in characteristic zero. They are defined in terms of multialternating polynomials non vanishing on them. Here we give a characterization of fundamental algebras in terms of representations of symmetric groups obtaining this way an equivalent definition. As an application we determine when a finitely generated Grassmann algebra is fundamental.
Symmetry-assisted adversaries for quantum state generation
2011
We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the $backslash$sc Graph Isomorphism problem. We show that for the related problem of $backslash$sc Index Erasure our method leads to a lower bound of $backslash Omega(backslash sqrt N)$ which matches an upper bound obtained via reduction to quantum search on $N$ elements. This closes an open problem first raised by Shi [FOCS'02]. Our approach is …
Catalan and Schröder permutations sortable by two restricted stacks
2020
Abstract Pattern avoiding machines were introduced recently by Claesson, Cerbai and Ferrari as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns. Some of the obtained results have been further generalized to Cayley permutations by Cerbai, specialized to particular patterns by Defant and Zheng, or considered in the context of functions over the symmetric group by Berlow. In this work we study pattern avoiding machines where the first stack avoids a pair of patterns of length 3 and investigate those pairs for which sortable permutations are counted by the…
A matrix of combinatorial numbers related to the symmetric groups
1979
For permutation groups G of finite degree we define numbers t"B(G)=|G|^-^[email protected]?"R"@?"[email protected]?"1(1a"1(g))^b^"^i, where B=(b"1,...,b"1) is a tuple of non-negative integers and a"1(g) denotes the number of i cycles in the element g. We show that t"B(G) is the number of orbits of G, acting on a set @D"B(G) of tuples of matrices. In the case G=S"n we get a natural interpretation for combinatorial numbers connected with the Stiring numbers of the second kind.
Symmetric-group approach to the study of the traces ofp-order reduced-density operators and of products of these operators
1990
In this work we give the values of traces of p-order reduced-density operators. These traces are obtained by application of the spin functions and of the symmetric-group properties. The relations obtained here will allow an easy and fast evaluation of the high-order spin-adapted reduced Hamiltonian matrix elements and high-order Hamiltonian moments.
Restricting irreducible characters to Sylow 𝑝-subgroups
2018
We restrict irreducible characters of finite groups of degree divisible by p p to their Sylow p p -subgroups and study the number of linear constituents.