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Combinatorial Hopf algebras and K-homology of Grassmanians

- T. Lam, P. Pylyavskyy
- Mathematics
- 15 May 2007

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as K-theoretic… Expand

Promotion and cyclic sieving via webs

- T. Petersen, P. Pylyavskyy, B. Rhoades
- Mathematics
- 21 April 2008

We show that Schützenberger’s promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work… Expand

Laurent phenomenon algebras

- T. Lam, P. Pylyavskyy
- Mathematics
- 12 June 2012

We generalize Fomin and Zelevinsky's cluster algebras by allowing exchange polynomials to be arbitrary irreducible polynomials, rather than binomials.

Tensor diagrams and cluster algebras

- S. Fomin, P. Pylyavskyy
- Mathematics
- 5 October 2012

The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V… Expand

A family of bijections between G-parking functions and spanning trees

- Denis Chebikin, P. Pylyavskyy
- Mathematics, Computer Science
- J. Comb. Theory, Ser. A
- 22 July 2003

For a directed graph G on vertices {0, 1, ..., n}, a G-parking function is an n-tuple (b1,...,bn) of non-negative integers such that, for every non-empty subset U ⊆ {1,...,n}, there exists a vertex j… Expand

Electrical networks and Lie theory

- T. Lam, P. Pylyavskyy
- Mathematics
- 17 March 2011

We introduce a new class of "electrical" Lie groups. These Lie groups, or more precisely their nonnegative parts, act on the space of planar electrical networks via combinatorial operations… Expand

Total positivity in loop groups I: whirls and curls

- T. Lam, P. Pylyavskyy
- Mathematics
- 4 December 2008

This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group… Expand

A non-crossing standard monomial theory

- T. Petersen, P. Pylyavskyy, David E Speyer
- Mathematics
- 11 June 2008

The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand-Tsetlin patterns and develop… Expand

Inverse Problem in Cylindrical Electrical Networks

- T. Lam, P. Pylyavskyy
- Mathematics, Computer Science
- SIAM J. Appl. Math.
- 26 April 2011

In this paper we study the inverse Dirichlet-to-Neumann problem for certain cylindrical electrical networks. We define and study a birational transformation acting on cylindrical electrical networks… Expand

Dual Filtered Graphs

- Rebecca Patrias, P. Pylyavskyy
- Mathematics
- 28 October 2014

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… Expand