Expand all Collapse all | Results 1 - 3 of 3 |
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 2001 (vol 53 pp. 470)
Hyperbolic Polynomials and Convex Analysis A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to
a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$
has all real roots for all vectors $x$. Motivated by partial
differential equations, G{\aa}rding proved in 1951 that the largest
such root is a convex function of $x$, and showed various ways of
constructing new hyperbolic polynomials. We present a powerful new
such construction, and use it to generalize G{\aa}rding's result to
arbitrary symmetric functions of the roots. Many classical and recent
inequalities follow easily. We develop various convex-analytic tools
for such symmetric functions, of interest in interior-point methods
for optimization problems over related cones.
Keywords:convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interior-point method, semidefinite program, singular value, symmetric function Categories:90C25, 15A45, 52A41 |