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Dual filtered graphs. (English) Zbl 1397.05202
Summary: We define a \(K\)-theoretic analogue of Fomin’s dual graded graphs, which we call dual filtered graphs. The key formula in the definition is \(DU-UD= D + I\). Our major examples are \(K\)-theoretic analogues of Young’s lattice, of shifted Young’s lattice, and of the Young-Fibonacci lattice. We suggest notions of tableaux, insertion algorithms, and growth rules whenever such objects are not already present in the literature. We also provide a large number of other examples. Most of our examples arise via two constructions, which we call the Pieri construction and the Möbius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described in [N. Bergeron et al., Algebr. Represent. Theory 15, No. 4, 675–696 (2012; Zbl 1281.16036); J. Nzeutchap “Dual graded graphs and Fomin’s \(r\)-correspondences associated to the Hopf algebras of planar binary trees, quasi-symmetric functions and noncommutative symmetric functions”, in: Proceedings of the 18th international conference on formal power series and algebraic combinatorics, FPSAC 2006, San Diego, CA, USA, June 19–23, 2006. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). 13 p. (2006), http://igm.univ-mlv.fr/~fpsac/FPSAC06/SITE06/papers/53.pdf]. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.

MSC:
05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
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