Cylindric plane partitions, Lambda determinants, Commutants in semicircular systems

TL;DR

This thesis explores three areas: cylindric plane partitions with new combinatorial proofs, a multi-parameter generalization of lambda-determinants, and properties of commutators in semicircular systems, advancing understanding in algebraic combinatorics and operator algebras.

Contribution

It provides a new bijective proof and a $(q,t)$-analog of Borodin's identity for cylindric plane partitions, a generalization of lambda-determinants, and insights into the structure of semicircular systems.

Findings

A bijective proof of Borodin's identity for cylindric plane partitions.

A $(q,t)$-analog of Borodin's identity with a combinatorial interpretation.

A projection matrix with a self-similar structure in semicircular systems.

Abstract

This thesis is divided into three parts. The first part deals with cylindric plane partitions. The second with lambda-determinants and the third with commutators in semi-circular systems. For more detailed abstract please see inside. Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. The first result of section one is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result is an explicit combinatorial interpretation of the Macdonald weight occurring in the $(q,t)$-analog using the non-intersecting lattice path model for cylindric plane partitions. Alternating sign matrices were discovered by Robbins and Rumsey whilst studying $\lambda$-determinants. In the second part of this thesis we prove a multi-parameter generalization of the $\lambda$-determinant, generalizing a recent result by di Francesco. Like the original $\lambda$-determinant, our formula exhibits the Laurent phenomenon. Semicircular systems were first introduced by Voiculescu as a part of his study of von Neumann algebras. In the third part of this thesis we study certain commutator subalgebras of the semicircular system. We find a projection matrix with an interesting self-similar structure. Making use of our projection formula we give an alternative, elementary proof that the semicircular system is a factor.