Expand all Collapse all | Results 1 - 8 of 8 |
1. CJM 2013 (vol 66 pp. 525)
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)
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)
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)
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)
Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of the complex reflection groups $G(r,p,n)$ |
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)
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)
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)
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 |