Crystals and total positivity on orientable surfaces
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- 作者：Thomas Lam (1)
Pavlo Pylyavskyy (2) - 关键词：Networks on surfaces ; Boundary measurements ; Geometric crystals ; Total positivity ; Loop groups ; 05C10 ; 05E10 ; 17B67 ; 22E65 ; 15B48
- 刊名：Selecta Mathematica, New Series
- 出版年：2013
- 出版时间：March 2013
- 年：2013
- 卷：19
- 期：1
- 页码：173-235
- 全文大小：1952KB
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Pavlo Pylyavskyy (2)
1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA
2. Department of Mathematics, University of Minnesota, Minneapolis, MN, 55414, USA - ISSN：1420-9020
文摘
We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated. We describe in terms of our model the crystal structure and R-matrix of the affine geometric crystal of products of symmetric and dual symmetric powers of type A. Local realizations of the R-matrix and crystal actions are used to construct a double affine geometric crystal on a torus, generalizing the commutation result of Kajiwara et?al. (Lett Math Phys, 60(3):211-19, 2002) and an observation of Berenstein and Kazhdan (MSJ Mem, 17:1-, 2007). We show that our model on a cylinder gives a decomposition and parametrization of the totally non-negative part of the rational unipotent loop group of GL n .