Algebra & Number Theory

Network parametrizations for the Grassmannian

Kelli Talaska and Lauren Williams

Full-text: Open access

Abstract

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 D of the Grassmannian are in bijection with certain tableaux D called Go-diagrams, and each component is isomorphic to (K)a×(K)b for some nonnegative integers a and b.

Our main result is an explicit parametrization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram D we construct a weighted network ND and its weight matrix WD, whose entries enumerate directed paths in ND. By letting the weights in the network vary over K or K as appropriate, one gets a parametrization of the Deodhar component D. One application of such a parametrization is that one may immediately determine which Plücker coordinates are vanishing and nonvanishing, by using the Lindström–Gessel–Viennot lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plücker coordinates. A main tool for us is the work of Marsh and Rietsch [Represent. Theory 8 (2004), 212–242] on Deodhar components in the flag variety.

Article information

Source
Algebra Number Theory, Volume 7, Number 9 (2013), 2275-2311.

Dates
Received: 19 October 2012
Revised: 15 March 2013
Accepted: 24 March 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730093

Digital Object Identifier
doi:10.2140/ant.2013.7.2275

Mathematical Reviews number (MathSciNet)
MR3152014

Zentralblatt MATH identifier
1281.14042

Subjects
Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]
Secondary: 05E10: Combinatorial aspects of representation theory [See also 20C30]

Keywords
Grassmannian network total positivity Deodhar decomposition

Citation

Talaska, Kelli; Williams, Lauren. Network parametrizations for the Grassmannian. Algebra Number Theory 7 (2013), no. 9, 2275--2311. doi:10.2140/ant.2013.7.2275. https://projecteuclid.org/euclid.ant/1513730093


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