Dual filtered graphs

Patrias, Rebecca; Pylyavskyy, Pavlo

Algebraic Combinatorics, Tome 1 (2018) no. 4, pp. 441-500.

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. (See the table below.) 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 [, , ]. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.

Classification : 05E99, 05E05
Mots clés : dual graded graphs, insertion algorithms, $K$-theory, symmetric functions

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