Lattice Diagram Polynomials and Extended Pieri Rules

Abstract

The lattice cell in the ${i+1}^{st}$ row and ${j+1}^{st}$ column of the positive quadrant of the plane is denoted $(i,j)$. If $\mu$ is a partition of $n+1$, we denote by $\mu /ij$ the diagram obtained by removing the cell $(i,j)$ from the (French) Ferrers diagram of $\mu$. We set ${\Delta }_{\mu /ij}=\det \Vert x_i^{p_j}{y}_i^{q_j}{\Vert }_{i,j=1}^n$, where $(p_1,q_1),...,(p_n,q_n)$ are the cells of $\mu /ij$, and let ${\mathbf{M}}_{\mu /ij}$ be the linear span of the partial derivatives of ${\Delta }_{\mu /ij}$. The bihomogeneity of ${\Delta }_{\mu /ij}$ and its alternating nature under the diagonal action of $S_n$ gives ${\mathbf{M}}_{\mu /ij}$ the structure of a bigraded $S_n$-module. We conjecture that ${\mathbf{M}}_{\mu /ij}$ is always a direct sum of $k$ left regular representations of $S_n$, where $k$ is the number of cells that are weakly north and east of $(i,j)$ in $\mu$. We also make a number of conjectures describing the precise nature of the bivariate Frobenius characteristic of ${\mathbf{M}}_{\mu /ij}$ in terms of the theory of Macdonald polynomials. On the validity of these conjectures, we derive a number of surprising identities. In particular, we obtain a representation theoretical interpretation of the coefficients appearing in some Macdonald Pieri Rules.

77 pages, TeX

References in corpus (3)

• Difference equations and symmetric polynomials defined by their zeros
• Symmetric and non-symmetric quantum Capelli polynomials
• Integrality of two variable Kostka functions

Cited by in corpus (10)

• Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
• Catalan paths, Quasi-symmetric functions and Super-Harmonic Spaces
• Vanishing ideals of Lattice Diagram determinants
• Diagonal Temperley-Lieb Invariants and Harmonics
• Lattice Diagram polynomials in one set of variables
• (GL_k x S_n)-modules and nabla of hook-indexed Schur functions
• Nested quantum Dyck paths and nabla(s_lambda)
• Schur Partial Derivative Operators
• A Survey of $q$-Whittaker polynomials
• The Combinatorics of the Garsia-Haiman Modules for Hook Shapes