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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 $mathfrak{gl}_{m|n}$. We discuss representation theoretic interpretation of our formulas using $q$-deformed $W$-algebra $mathfrak{gl}_{m|n}$.
The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is pr
For any triple $(i,a,mu)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $mu$, we define a quiver current $H^{(i,a)}_mu$ acting on the tensor power $Lambda^Q$ of symmetric functions over the vertices
We present an algorithm for approximating linear categories of partitions (of sets). We report on concrete computer experiments based on this algorithm which we used to obtain first examples of so-called non-easy linear categories of partitions. All
For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in ter
Plane partitions have been widely studied in Mathematics since MacMahon. See, for example, the works by Andrews, Macdonald, Stanley, Sagan and Krattenthaler. The Schur process approach, introduced by Okounkov and Reshetikhin, and further developed by