Network parameterizations for the Grassmannian

Talaska, Kelli ; Williams, Lauren

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013).

Résumé

Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation, as shown by Kodama and the second author. Deodhar components $S_D$ of the Grassmannian are in bijection with certain tableaux $D$ called $\textit{Go-diagrams}$, and each component is isomorphic to $(\mathbb{K}^*)^a ×\mathbb{K})^b$ for some non-negative integers $a$ and $b$. Our main result is an explicit parameterization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram $D$ we construct a weighted network $N_D$ and its $\textit{weight matrix}$ $W_D$, whose entries enumerate directed paths in $N_D$. By letting the weights in the network vary over $\mathbb{K}$ or $\mathbb{K} ^*$ as appropriate, one gets a parametrization of the Deodhar component $S_D$. One application of such a parametrization is that one may immediately determine which Plücker coordinates are vanishing and nonvanishing, by using the Lindstrom-Gessel-Viennot Lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plücker coordinates.

Détail

DOI : 10.46298/dmtcs.12793