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, the binary tree, and the graph determined by the Poirier-Reutenauer Hopf algebra. 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 in Bergeron-Lam-Li, Nzeutchap, and Lam-Shimozono. The Mobius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.
Highlights
Fomin’s dual graded graphs [10], as well as their predecessors - Stanley’s differential posets [26], were invented as a tool to better understand the Robinson–Schensted insertion algorithm
We show how the Pieri construction applied to the ring generated by Grothendieck polynomials yields the same result as the Möbius construction applied to Young’s lattice
The Hopf algebra of quasisymmetric functions and its K-theoretic analogue provide another instance of the Möbius via Pieri phenomenon
Summary
Fomin’s dual graded graphs [10], as well as their predecessors - Stanley’s differential posets [26], were invented as a tool to better understand the Robinson–Schensted insertion algorithm. Graded graphs G1 and G2 with a common vertex set and rank function are said to be dual if DU − UD = I, where I is an identity operator acting on KP. One would expect that variations of the Weyl algebra would correspond to some variations of the theory of dual graded graphs. This is not the case for K-theoretic analogues of the Poirier–Reutenauer and Malvenuto– Reutenauer Hopf algebras, as described in Sections 7.2, 7.3 To put it the numbers of basis elements for the filtered components of A and Aare distinct in those cases, there is no hope to obtain corresponding graphs one from the other via Möbius construction. The Hopf algebra of quasisymmetric functions and its K-theoretic analogue provide another instance of the Möbius via Pieri phenomenon. We formulate and prove a K-theoretic analogue of the Frobenius–Young identity
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