Lattice Diagram Polynomials and Extended Pieri Rules

Details Lattice Diagram Polynomials and Extended Pieri Rules Lattice Diagram Polynomials and Extended Pieri Rules

1998-09-22

arxiv.org/abs/math/9809126v1

Adv. Math. 142 (1999), no. 2, 244--334.

Mathematics

Combinatorics

77 pages, TeX

Scientific paper

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 $\Delta_{\mu/ij}=\det \| x_i^{p_j}y_i^{q_j} \|_{i,j=1}^n$, where $(p_1,q_1),... ,(p_n,q_n)$ are the cells of $\mu/ij$, and let ${\bf M}_{\mu/ij}$ 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 ${\bf M}_{\mu/ij}$ the structure of a bigraded $S_n$-module. We conjecture that ${\bf M}_{\mu/ij}$ 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 ${\bf M}_{\mu/ij}$ 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.

Affiliated with

Also associated with

No associations

Rating

Lattice Diagram Polynomials and Extended Pieri Rules does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Lattice Diagram Polynomials and Extended Pieri Rules, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Lattice Diagram Polynomials and Extended Pieri Rules will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-360895