Plane partitions with a "pit": generating functions and representation theory
Abstract
We study plane partitions satisfying condition $a_{n+1,m+1}=0$ (this condition is called "pit") and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal $\mathfrak{gl}_1$ algebra, therefore our formulas can be interpreted as the characters of these representations. The resulting formulas resemble formulas for characters of tensor representations of Lie superalgebra $\mathfrak{gl}_{mn}$. We discuss representation theoretic interpretation of our formulas using $q$deformed $W$algebra $\mathfrak{gl}_{mn}$.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 DOI:
 10.48550/arXiv.1512.08779
 arXiv:
 arXiv:1512.08779
 Bibcode:
 2015arXiv151208779B
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Quantum Algebra;
 Mathematics  Representation Theory
 EPrint:
 30 pages, v2. 32 pages, new subsection 4.4 included, v3 36 pages, many corrections, references added, v4 38 pages many corrections, references added, version to appear in Sel. Math. New Ser. (2018)