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Search: MSC category 05E05 ( Symmetric functions and generalizations )

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1. CJM 2013 (vol 66 pp. 525)

Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike
A Lift of the Schur and Hall-Littlewood Bases to Non-commutative Symmetric Functions
We introduce a new basis of the algebra of non-commutative symmetric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.

Keywords:Hall-Littlewood polynomial, symmetric function, quasisymmetric function, tableau
Category:05E05

2. CJM 2011 (vol 64 pp. 822)

Haglund, J.; Morse, J.; Zabrocki, M.
A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path
We introduce a $q,t$-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory $\nabla$ operator applied to a Hall--Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the ``shuffle conjecture" (Duke J. Math. $\mathbf {126}$ ($2005$), 195-232) for $\nabla e_n[X]$. We bring to light that certain generalized Hall--Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of $q,t$-Catalan sequences, and we prove a number of identities involving these functions.

Keywords:Dyck Paths, Parking functions, Hall--Littlewood symmetric functions
Categories:05E05, 33D52

3. CJM 2007 (vol 59 pp. 36)

Develin, Mike; Martin, Jeremy L.; Reiner, Victor
Classification of Ding's Schubert Varieties: Finer Rook Equivalence
K.~Ding studied a class of Schubert varieties $X_\lambda$ in type A partial flag manifolds, indexed by integer partitions $\lambda$ and in bijection with dominant permutations. He observed that the Schubert cell structure of $X_\lambda$ is indexed by maximal rook placements on the Ferrers board $B_\lambda$, and that the integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$ satisfy the combinatorial condition of \emph{rook-equivalence}. We classify the varieties $X_\lambda$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

Keywords:Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotence
Categories:14M15, 05E05

4. CJM 2002 (vol 54 pp. 239)

Cartwright, Donald I.; Steger, Tim
Elementary Symmetric Polynomials in Numbers of Modulus $1$
We describe the set of numbers $\sigma_k(z_1,\ldots,z_{n+1})$, where $z_1,\ldots,z_{n+1}$ are complex numbers of modulus $1$ for which $z_1z_2\cdots z_{n+1}=1$, and $\sigma_k$ denotes the $k$-th elementary symmetric polynomial. Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus. Another application is the calculation of the spectrum of certain adjacency operators arising naturally on a building of type ${\tilde A}_n$.

Categories:05E05, 33C45, 30C15, 51E24

5. CJM 1998 (vol 50 pp. 167)

Halverson, Tom; Ram, Arun
Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of the complex reflection groups $G(r,p,n)$
Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,p,n)$ were constructed recently in the work of Ariki and Koike~\cite{AK}, Brou\'e and Malle \cite{BM}, and Ariki~\cite{Ari}. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper ~\cite{HR} in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene~\cite{Gre}, who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of $G(r,p,n)$ given by Ariki and Koike~\cite{AK} and Ariki ~\cite{Ari}.

Categories:20C05, 05E05

6. CJM 1997 (vol 49 pp. 883)

Okounkov, Andrei
Proof of a conjecture of Goulden and Jackson
We prove an integration formula involving Jack polynomials conjectured by I.~P.~Goulden and D.~M.~Jackson in connection with enumeration of maps in surfaces.

Categories:05E05, 43A85, 57M15

7. CJM 1997 (vol 49 pp. 865)

Goulden, I. P.; Jackson, D. M.
Maps in locally orientable surfaces and integrals over real symmetric surfaces
The genus series for maps is the generating series for the number of rooted maps with a given number of vertices and faces of each degree, and a given number of edges. It captures topological information about surfaces, and appears in questions arising in statistical mechanics, topology, group rings, and certain aspects of free probability theory. An expression has been given previously for the genus series for maps in locally orientable surfaces in terms of zonal polynomials. The purpose of this paper is to derive an integral representation for the genus series. We then show how this can be used in conjunction with integration techniques to determine the genus series for monopoles in locally orientable surfaces. This complements the analogous result for monopoles in orientable surfaces previously obtained by Harer and Zagier. A conjecture, subsequently proved by Okounkov, is given for the evaluation of an expectation operator acting on the Jack symmetric function. It specialises to known results for Schur functions and zonal polynomials.

Categories:05C30, 05A15, 05E05, 15A52

8. CJM 1997 (vol 49 pp. 263)

Hamel, A. M.
Determinantal forms for symplectic and orthogonal Schur functions
Symplectic and orthogonal Schur functions can be defined combinatorially in a manner similar to the classical Schur functions. This paper demonstrates that they can also be expressed as determinants. These determinants are generated using planar decompositions of tableaux into strips and the equivalence of these determinants to symplectic or orthogonal Schur functions is established by Gessel-Viennot lattice path techniques. Results for rational (also called {\it composite}) Schur functions are also obtained.

Categories:05E05, 05E10, 20C33

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