Quasi-commuting families of quantum minors

By Joshua S. Scott. 2000

54 citations recorded by the source corpus.

In [B. Leclerc, A. Zelevinsky, Quasicommuting families of quantum Plucker coordinates, in: Kirillov's Seminar on Representation Theory, in: Amer. Math. Soc. Transl. (2), vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 85–108], a combinatorial criterion for quasi-commutativity is established for pairs of quantum Plucker coordinates in the quantized coordinate algebra Cq[F] of the flag variety of type A. This paper attempts to generalize these results by producing necessary and sufficient conditions for pairs of quantum minors in the quantized coordinate algebra Cq[Matk×m] to quasi-commute. In addition, we study the combinatorics of maximal (by inclusion) families of pairwise quasi-commuting quantum minors and pose relevant conjectures.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/267f8cc882edc1c2426f494de7dbf5d2e4a5f3fc

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Citations in the corpus

A categorification of Grassmannian cluster algebras (2013, background): “We thank Jeanne Scott, Andrei Zelevinsky, Bernard Leclerc, Jan Schröer, Michael Wemyss, Osamu Iyama, Matthew Pressland, Karin Baur and Robert Marsh for helpful discussions and input at various stages of this project.”
Total positivity, Grassmannians, and networks (2006, methodology): “Scott [ Sco1 , Sco2] used our alternating strand diagrams to study Leclerc-Zelevinsky’s [LZ] quasi-commuting families of quantum minors and cluster algebra on the Grassmannian.”
Grassmannians and Cluster Algebras (2003, methodology): “With this in mind, [9] combinatorially described all pairs of Plücker coordinates in the Grassmannian G(k, n) which quasi-commute, while [13] proved that the set of monomials consisting of products of pairwise quasi-commuting Plücker coordinates is in fact a basis for the coordinate ring C[G(2, n)].”
Weak separation and plabic graphs (2011, background): “e assign the decoratedstrand permutation π: G = (πG,col) to each reduced plabic graph G. We will label the faces of a reduced plabic graph with subsets of [n]; this construction was ﬁrst published in [10]. By condition 2 each strand divides the disk into two parts. For each face F we label that F wi”