Positivity for special cases of -Kostka coefficients and standard tableaux statistics.
Positivity in coefficient-free rank two cluster algebras.
Positivity of Schur function expansions of Thom polynomials
Combining the approach to Thom polynomials via classifying spaces of singularities with the Fulton-Lazarsfeld theory of cone classes and positive polynomials for ample vector bundles, we show that the coefficients of the Schur function expansions of the Thom polynomials of stable singularities are nonnegative with positive sum.
Positivity of the -system cluster algebra.
Positivity of Thom polynomials II: the Lagrange singularities
We study Thom polynomials associated with Lagrange singularities. We expand them in the basis of Q̃-functions. This basis plays a key role in the Schubert calculus of isotropic Grassmannians. We prove that the Q̃-function expansions of the Thom polynomials of Lagrange singularities always have nonnegative coefficients. This is an analog of a result on the Thom polynomials of mapping singularities and Schur S-functions, established formerly by the last two authors.
Pretty cleanness and filter-regular sequences
Let be a field and . Let be a monomial ideal of and be monomials in . We prove that if form a filter-regular sequence on , then is pretty clean if and only if is pretty clean. Also, we show that if form a filter-regular sequence on , then Stanley’s conjecture is true for if and only if it is true for . Finally, we prove that if is a minimal set of generators for which form either a -sequence, proper sequence or strong -sequence (with respect to the reverse lexicographic...
Problems in algebraic combinatorics.
Product and puzzle formulae for Belkale-Kumar coefficients.
Products of factorial Schur functions.
Projective representations of generalized symmetric groups.
Promotion and evacuation on standard Young tableaux of rectangle and staircase shape.
Promotion operator on rigged configurations of type .
Pseudodual grids and extensions of generalized quadrangles.
-identities related to overpartitions and divisor functions.
-Selberg integrals and Macdonald polynomials
Quadratic Equations over Finite Fields and Class Numbers of Real Quadratic Fields.
Quasigroup automorphisms and symmetric group characters
The automorphisms of a quasigroup or Latin square are permutations of the set of entries of the square, and thus belong to conjugacy classes in symmetric groups. These conjugacy classes may be recognized as being annihilated by symmetric group class functions that belong to a -ideal of the special -ring of symmetric group class functions.
Quelques remarques sur une Construction de Schensted.
Quelques remarques sur une construction de Schensted
Radicals of symmetric cellular algebras
For a symmetric cellular algebra, we study properties of the dual basis of a cellular basis first. Then a nilpotent ideal is constructed. The ideal connects the radicals of cell modules with the radical of the algebra. It also yields some information on the dimensions of simple modules. As a by-product, we obtain some equivalent conditions for a finite-dimensional symmetric cellular algebra to be semisimple.