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193
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 ..."
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Cited by 189 (24 self)
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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 ..."
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Cited by 69 (4 self)
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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 ..."
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Cited by 67 (0 self)
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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 ..."
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Cited by 54 (26 self)
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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 ..."
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Cited by 44 (13 self)
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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
 EUROP. J. COMBIN
, 2001
"... 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. ..."
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Cited by 35 (7 self)
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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 ..."
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Cited by 29 (5 self)
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this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions
Descent representations and multivariate statistics
 Trans. Amer. Math. Soc
"... Abstract. Combinatorial identities on Weyl groups of types A and B are derived from special bases of the corresponding coinvariant algebras. Using the GarsiaStanton descent basis of the coinvariant algebra of type A we give a new construction of the Solomon descent representations. An extension of ..."
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Cited by 22 (8 self)
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Abstract. Combinatorial identities on Weyl groups of types A and B are derived from special bases of the corresponding coinvariant algebras. Using the GarsiaStanton descent basis of the coinvariant algebra of type A we give a new construction of the Solomon descent representations. An extension of the descent basis to type B, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine wellknown decomposition rules of the coinvariant algebra and to generalize various identities. 1.