Dual Filtered Graphs

TL;DR

This paper introduces dual filtered graphs as a K-theoretic analogue of dual graded graphs, providing new examples, constructions, and algorithms related to combinatorial and algebraic structures.

Contribution

It defines dual filtered graphs with a key formula, presents new examples including K-theoretic analogues of classical lattices, and introduces two constructions for generating these graphs.

Findings

Introduces dual filtered graphs with DU-UD= D + I formula

Provides K-theoretic analogues of classical combinatorial lattices

Develops the Pieri and Mobius constructions for these graphs

Abstract

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 Mobius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described by Bergeron-Lam-Li, Nzeutchap, and Lam-Shimizono. The Mobius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.