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.

scholarly journals The center of the wreath product of symmetric group algebras

2021 ◽  
Vol 31 (2) ◽  
pp. 302-322
Author(s):  
O. Tout ◽  
Keyword(s):  
Symmetric Group ◽  
Recent Result ◽  
Wreath Product ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Group Algebras ◽  
Hyperoctahedral Group ◽  
Symmetric Group Algebras

We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric group algebras. This allows us to recover an old result of Farahat and Higman about the polynomiality of the structure coefficients of the center of the symmetric group algebra and to generalize our recent result about the polynomiality property of the structure coefficients of the center of the hyperoctahedral group algebra.

ON SOME ALGEBRAIC AND COMBINATORIAL PROPERTIES OF DUNKL ELEMENTS

2012 ◽  
Vol 26 (27n28) ◽  
pp. 1243012 ◽  
Author(s):  
ANATOL N. KIRILLOV
Keyword(s):  
Symmetric Group ◽  
Group Ring ◽  
Group Algebra ◽  
Schubert Calculus ◽  
The Other ◽  
Flag Varieties ◽  
Ehrhart Polynomial ◽  
Quadratic Algebras ◽  
Combinatorial Properties ◽  
Quantum Version

We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements and Schubert calculus, in Advances in Geometry (eds. J.-S. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, Boston, 1995), pp. 147–182, A. Postnikov, On a quantum version of Pieri's formula, in Advances in Geometry (eds. J.-S. Brylinski, R. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, 1995), pp. 371–383 and A. N. Kirillov and T. Maenor, A Note on Quantum K-Theory of Flag Varieties, preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [E. Mukhin, V. Tarasov and A. Varchenko, Bethe Subalgebras of the Group Algebra of the Symmetric Group, preprint arXiv:1004.4248]. Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang–Baxter algebra in a connection with the values of the β-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan–Robbins polytope.

scholarly journals On semisymmetric connection

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1179-1184
Author(s):  
Ana Velimirovic ◽  
Milan Zlatanovic
Keyword(s):  
Symmetric Space ◽  
Curvature Tensor ◽  
The Other ◽  
Linear Combinations ◽  
Covariant Derivatives ◽  
Linearly Independent ◽  
Curvature Tensors

Using the non-symmetry of a connection, it is possible to introduce four types of covariant derivatives. Based on these derivatives, several types of Ricci?s identities and twelve curvature tensors are obtained. Five of them are linearly independent but the other curvature tensors can be expressed as linear combinations of these five linearly independent curvature tensors and the curvature tensor of the corresponding associated symmetric space. The semisymmetric connection is defined and the properties of two of the five independent curvature tensors are analyzed. In the same manner, the properties for three others curvature tensors may be derived.

scholarly journals On Division Versus Saturation in Pseudo-Boolean Solving

2019 ◽  
Author(s):  
Stephan Gocht ◽  
Jakob Nordström ◽  
Amir Yehudayoff
Keyword(s):  
Linear Programming ◽  
Cutting Planes ◽  
The Other ◽  
Sat Solving ◽  
Linear Combinations ◽  
Division Rule ◽  
Clause Learning

The conflict-driven clause learning (CDCL) paradigm has revolutionized SAT solving over the last two decades. Extending this approach to pseudo-Boolean (PB) solvers doing 0-1 linear programming holds the promise of further exponential improvements in theory, but intriguingly such gains have not materialized in practice. Also intriguingly, most PB extensions of CDCL use not the division rule in cutting planes as defined in [Cook et al., '87] but instead the so-called saturation rule. To the best of our knowledge, there has been no study comparing the strengths of division and saturation in the context of conflict-driven PB learning, when all linear combinations of inequalities are required to cancel variables. We show that PB solvers with division instead of saturation can be exponentially stronger. In the other direction, we prove that simulating a single saturation step can require an exponential number of divisions. We also perform some experiments to see whether these phenomena can be observed in actual solvers. Our conclusion is that a careful combination of division and saturation seems to be crucial to harness more of the power of cutting planes.

Induced Representations and Invariants

1950 ◽  
Vol 2 ◽  
pp. 334-343 ◽  
Author(s):  
G. DE B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Explicit Formula ◽  
Linear Group ◽  
Schur Functions ◽  
The Other ◽  
Induced Representations ◽  
Direct Sum ◽  
Invariant Matrices ◽  
The One

1. Introduction. The problem of the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices is intimately associated with the representation theory of the full linear group on the one hand and with the representation theory of the symmetric group on the other. In a previous paper the author gave an explicit formula for this reduction in terms of characters of the symmetric group. Later J. A. Todd derived the same formula using Schur functions, i.e. characters of representations of the full linear 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 Linear combinations of prime powers in binary recurrence sequences

2017 ◽  
Vol 13 (02) ◽  
pp. 261-271 ◽  
Author(s):  
Csanád Bertók ◽  
Lajos Hajdu ◽  
István Pink ◽  
Zsolt Rábai
Keyword(s):  
Linear Combination ◽  
The Other ◽  
Linear Recurrence ◽  
Linear Combinations ◽  
Other Hand ◽  
Sums Of Powers ◽  
Recurrence Sequences

We give finiteness results concerning terms of linear recurrence sequences having a representation as a linear combination, with fixed coefficients, of powers of fixed primes. On one hand, under certain conditions, we give effective bounds for the terms of binary recurrence sequences with such a representation. On the other hand, in the case of some special binary recurrence sequences, all terms having a representation as sums of powers of [Formula: see text] and [Formula: see text] are explicitly determined.

ON THE CENTER OF THE MODULAR GROUP ALGEBRA OF A FINITE p-GROUP

2014 ◽  
Vol 13 (04) ◽  
pp. 1350127
Author(s):  
CZESŁAW BAGIŃSKI ◽  
JÁNOS KURDICS
Keyword(s):  
Group Algebra ◽  
Modular Group ◽  
Conjugacy Classes ◽  
Characteristic P ◽  
Cyclic Groups ◽  
Modular Group Algebra ◽  
Index 2 ◽  
Zero Multiplication

Let G be a finite nonabelian p-group and F a field of characteristic p and let [Formula: see text] be the subalgebra spanned by class sums [Formula: see text], where C runs over all conjugacy classes of noncentral elements of G. We show that all finite p-groups are subgroups and homomorphic images of p-groups for which [Formula: see text]. We also give the description of abelian-by-cyclic groups for which [Formula: see text] is an algebra with zero multiplication or is nil of index 2.

A pencil of four-nodal plane sextics

1981 ◽  
Vol 89 (3) ◽  
pp. 413-421 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Symmetric Group ◽  
The Other ◽  
Nodal Plane ◽  
Geometrical Properties ◽  
Index 2

Wiman, in 1895, found ((5), p. 208) an equation for a 4-nodal plane sextic W that admits a group S of 120 Cremona self-transformations; of these, 24 are projectivities, the other 96 quadratic transformations. S is isomorphic to the symmetric group of degree 5 and Wiman emphasizes that S does permute among themselves 5 pencils (4 pencils of lines and 1 of conics) and 5 nets (4 nets of conics and 1 of lines). But he gives no geometrical properties of W. The omission should be repaired because, as will be explained below, W can be uniquely determined by elementary geometrical conditions. Furthermore: W is only one, though admittedly the most interesting, of a whole pencil P of 4-nodal sextics; every member of P is invariant under S+, the icosahedral subgroup of index 2 in S, while the transformations in the coset S/S+ transpose the members of P in pairs save for two that they leave fixed, W being one of these. When the triangle of reference is the diagonal point triangle of the quadrangle of its nodes the form of W is (7.2) below. Wiman referred his curve to a different triangle.

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