scholarly journals A Combinatorial Note for Harmonic Tensors

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Zhankui Xiao
Keyword(s):  
Symmetric Group ◽  
Group Algebra

We give another characterization of the annihilator of the space of (dual) harmonic tensors in the group algebra of symmetric group.

On the Garsia Lie Idempotent

2005 ◽  
Vol 48 (3) ◽  
pp. 445-454 ◽  
Author(s):  
Frédéric Patras ◽  
Christophe Reutenauer ◽  
Manfred Schocker
Keyword(s):  
Symmetric Group ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Group Algebra ◽  
Orthogonal Projection ◽  
Gram Matrix ◽  
Free Associative Algebra ◽  
Free Lie Algebra ◽  
Rational Group ◽  
Lie Idempotent

AbstractThe orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group Sn, in each homogenous degree n. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of Sn−1. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in Sn.

scholarly journals A characterization of PGL (2,pn) by some irreducible complex character degrees

2016 ◽  
Vol 99 (113) ◽  
pp. 257-264 ◽  
Author(s):  
Somayeh Heydari ◽  
Neda Ahanjideh
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Cyclic Group ◽  
Group Algebra ◽  
Character Degrees ◽  
Complex Character ◽  
Complex Group ◽  
Complex Group Algebra ◽  
A Finite Group

For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that |G| = |PGL(2,p) |, p ? cd(G) and max(cd(G)) = p+1, then G ? PGL(2,p), SL(2, p) or PSL(2,p) x A, where A is a cyclic group of order (2, p-1). Also, we show that if G is a finite group with X1(G) = X1(PGL(2,pn)), then G ? PGL(2, pn). In particular, this implies that PGL(2, pn) is uniquely determined by the structure of its complex group algebra.

scholarly journals Structure Coefficients of the Hecke Algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$

10.37236/3592 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Omar Tout
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Main Tool ◽  
Hecke Algebra ◽  
Natural Analogue ◽  
New Class ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras ◽  
Combinatorial Algebra

The Hecke algebra of the pair $(\mathcal{S}_{2n},\mathcal{B}_n)$, where $\mathcal{B}_n$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial algebra which projects onto the Hecke algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$ for every $n$. To build it, by using partial bijections we introduce and study a new class of finite dimensional algebras.

A Generalization of the Young Diagram

1954 ◽  
Vol 6 ◽  
pp. 498-508 ◽  
Author(s):  
M. D. Burrow
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Young Diagram ◽  
Primitive Idempotents ◽  
Detailed Treatment

The method of A. Young for finding the set of primitive idempotents of the group algebra of the symmetric group is classical; it was first given by Frobenius (4) using results of Young (10 and 11). A concise account can be found in (9) and a very detailed treatment in (6).

Twisted Group Rings Whose Units Form an FC-Group

1995 ◽  
Vol 47 (2) ◽  
pp. 274-289
Author(s):  
Victor Bovdi
Keyword(s):  
Group Algebra ◽  
Group Algebras ◽  
Group Rings ◽  
Twisted Group Algebra ◽  
Group Of Units ◽  
Twisted Group

AbstractLet U(KλG) be the group of units of the infinite twisted group algebra KλG over a field K. We describe the FC-centre ΔU of U(KλG) and give a characterization of the groups G and fields K for which U(KλG) = ΔU. In the case of group algebras we obtain the Cliff-Sehgal-Zassenhaus theorem.

The Poset of Conjugacy Classes and Decomposition of Products in the Symmetric Group

1992 ◽  
Vol 35 (2) ◽  
pp. 152-160 ◽  
Author(s):  
François Bédard ◽  
Alain Goupil
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Conjugacy Classes ◽  
The Other ◽  
Linear Combinations ◽  
Graded Poset

AbstractThe action by multiplication of the class of transpositions of the symmetric group on the other conjugacy classes defines a graded poset as described by Birkhoff ([2]). In this paper, the edges of this poset are given a weight and the structure obtained is called the poset of conjugacy classes of the symmetric group. We use weights of chains in the posets to obtain new formulas for the decomposition of products of conjugacy classes of the symmetric group in its group algebra as linear combinations of conjugacy classes and we derive a new identity involving partitions of n.

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