Results 1  10
of
191
Oligomorphic permutation groups
 LONDON MATHEMATICAL SOCIETY STUDENT TEXTS
, 1999
"... A permutation group G (acting on a set Ω, usually infinite) is said to be oligomorphic if G has only finitely many orbits on Ω n (the set of ntuples of elements of Ω). Such groups have traditionally been linked with model theory and combinatorial enumeration; more recently their grouptheoretic pro ..."
Abstract

Cited by 185 (24 self)
 Add to MetaCart
A permutation group G (acting on a set Ω, usually infinite) is said to be oligomorphic if G has only finitely many orbits on Ω n (the set of ntuples of elements of Ω). Such groups have traditionally been linked with model theory and combinatorial enumeration; more recently their grouptheoretic properties have been studied, and links with graded algebras, Ramsey theory, topological dynamics, and other areas have emerged. This paper is a short summary of the subject, concentrating on the enumerative and algebraic aspects but with an account of grouptheoretic properties. The first section gives an introduction to permutation groups and to some of the more specific topics we require, and the second describes the links to model theory and enumeration. We give a spread of examples, describe results on the growth rate of the counting functions, discuss a graded algebra associated with an oligomorphic group, and finally discuss grouptheoretic properties such as simplicity, the small index property, and “almostfreeness”.
Quasishuffle products
 J. Algebraic Combin
"... Abstract. Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication ∗ on the set of noncommutative polynomials in A which we call a quasishuffle product; it can be viewed as a generalization of the shuffle product ..."
Abstract

Cited by 69 (4 self)
 Add to MetaCart
Abstract. Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication ∗ on the set of noncommutative polynomials in A which we call a quasishuffle product; it can be viewed as a generalization of the shuffle product x. We extend this commutative algebra structure to a Hopf algebra (A, ∗, ∆); in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasisymmetric functions. If rational coefficients are allowed, the quasishuffle product is in fact no more general than the shuffle product; we give an isomorphism exp of the shuffle Hopf algebra (A,x,∆) onto (A, ∗, ∆). Both the set L of Lyndon words on A and their images {exp(w)  w ∈ L} freely generate the algebra (A, ∗). We also consider the graded dual of (A, ∗,∆). We define a deformation ∗q of ∗ that coincides with ∗ when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasisymmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.
Kontsevich’s universal formula for deformation quantization
 and the CBH formula, I, math.QA/9811174
"... Abstract. We relate a universal formula for the deformation quantization of Poisson structures (⋆products) on R d proposed by Maxim Kontsevich to the CampbellBakerHausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for su ..."
Abstract

Cited by 67 (0 self)
 Add to MetaCart
Abstract. We relate a universal formula for the deformation quantization of Poisson structures (⋆products) on R d proposed by Maxim Kontsevich to the CampbellBakerHausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for such formulae. For the dual of a Lie algebra, the ⋆product given by the universal enveloping algebra via symmetrization is shown to be of this type. In fact this ⋆product is essentially given by the CampbellBakerHausdorff (CBH) formula. We call this the CBHquantization. Next we limn Kontsevich’s construction using a graphical representation for differential calculus. We outline a structure theory for the weighted graphs which encode bidifferential operators in his formula and compute certain weights. We then establish that the Kontsevich and CBH quantizations are identical for the duals of nilpotent Lie algebras. Consequently part of Kontsevich’s ⋆product is determined by the CBH formula. Working the other way, we have a graphical encoding for the
Noncommutative Symmetric Functions II: Transformations Of Alphabets
 J. of Alg. and Comput
"... Noncommutative analogues of classical operations on symmetric functions are investigated, and applied to the description of idempotents and nilpotents in descent algebras. Its is shown that any sequence of Lie idempotents (one in each descent algebra) gives rise to a complete set of indecomposable o ..."
Abstract

Cited by 54 (26 self)
 Add to MetaCart
Noncommutative analogues of classical operations on symmetric functions are investigated, and applied to the description of idempotents and nilpotents in descent algebras. Its is shown that any sequence of Lie idempotents (one in each descent algebra) gives rise to a complete set of indecomposable orthogonal idempotents of each descent algebra, and various deformations of the classical sequences of Lie idempotents are obtained. In particular, we obtain several qanalogues of the Eulerian idempotents and of the GarsiaReutenauer idempotents.
Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at q = 0
 J. ALG. COMB
, 1997
"... We present representation theoretical interpretations of quasisymmetric functions and noncommutative symmetric functions in terms of quantum linear groups and Hecke algebras at q = 0. We obtain in this way a noncommutative realization of quasisymmetric functions analogous to the plactic symmetric ..."
Abstract

Cited by 44 (13 self)
 Add to MetaCart
We present representation theoretical interpretations of quasisymmetric functions and noncommutative symmetric functions in terms of quantum linear groups and Hecke algebras at q = 0. We obtain in this way a noncommutative realization of quasisymmetric functions analogous to the plactic symmetric functions of Lascoux and Schützenberger. The generic case leads to a notion of quantum Schur functions.
The Flag Major Index and Group Actions on Polynomial Rings
, 2000
"... A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type B, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra. ..."
Abstract

Cited by 35 (7 self)
 Add to MetaCart
A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type B, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.
Parking Functions and Noncrossing Partitions
 Electronic J. Combinatorics
, 1997
"... this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions ..."
Abstract

Cited by 29 (5 self)
 Add to MetaCart
this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions
Noncommutative Symmetric Functions III: Deformations Of Cauchy And Convolution Algebras
"... This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple qanalogue of the shuffle product, which has unexpec ..."
Abstract

Cited by 22 (8 self)
 Add to MetaCart
This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple qanalogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in theoretical physics (the quon algebra).