Related papers: A basis of the basic $sl(3,C)\sptilde$-module
For $k\geq i\geq 1$, let $B_{k,i}(n)$ denote the number of partitions of $n$ such that part 1 appears at most $i-1$ times, two consecutive integers l and $l+1$ appear at most $k-1$ times and if l and $l+1$ appear exactly $k-1$ times then…
We give a descent monomial basis of $\Delta$-Springer modules $R_{n,\lambda,s}$, first defined by Griffin. Our construction simultaneously generalizes the descent basis for the Garsia-Procesi module $R_\lambda$ studied by Carlsson-Chou and…
Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum $q$-series. Equipped with such a combinatorial set-up,…
In this article, we study the arithmetic properties of the partition function $p_8(n)$, the number of 8-colour partitions of $n$. We prove several Ramanujan type congruences modulo higher powers of 2 for the function $p_8(n)$ by finding…
We consider simple modules for a Hecke algebra with a parameter of quantum characteristic $e$. Equivalently, we consider simple modules $D^{\lambda}$, labelled by $e$-restricted partitions $\lambda$ of $n$, for a cyclotomic KLR algebra…
We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the…
Noncommutative invariant theory is a generalization of the classical invariant theory of the action of $SL(2,\IC)$ on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain…
A cuspidal system for an affine Khovanov-Lauda-Rouquier algerba $R_\al$ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\al$ up to the so-called imaginary modules. We make a conjecture on…
In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that L decomposes as a module for that sl3 subalgebra into copies of the adjoint module, the natural 3-dimensional module and its dual, and the…
In his recent work, Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions.…
The color structure needed for resummation of all colored 2 -> 3 processes is calculated using multiplet inspired s-channel bases. In this way the resulting matrices, describing the color structure, are guaranteed to obey simplifying…
We examine the cell modules for the category of type An webs and their natural cellular forms. We modify the bases of these modules, as described by Elias, to obtain an orthogonal basis of each cell module. Hence, we calculate the…
Colored planar rook algebra is a semigroup algebra in which the basis element has a diagrammatic description. The category of finite dimensional modules over this algebra is completely reducible and suitable functors are defined on this…
Inspired by the work of S. Ramanujan, many people have studied generalized modular equations and the numerous identities found by Ramanujan. These identities known as modular equations can be transformed into polynomial equations. There is…
Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…
We consider the homogeneous components U_r of the map on R = k[x,y,z]/(x^A, y^B, z^C) that multiplies by x + y + z. We prove a relationship between the Smith normal forms of submatrices of an arbitrary Toeplitz matrix using Schur…
We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new partition identities related to the…
We classify irreducible SL(2,K)-modules of low Morley rank (at most 4.rk(K)) as a first step towards a more general conjecture.
We prove the existence of a new algorithm for 3-sphere recognition based on Groebner basis methods applied to the variety of $\text{\em SL}(2,\C)$-representation of the fundamental group. An essential input is a recent result of the second…
Extensions of a set partition obtained by imposing bounds on the size of the parts and the coloring of some of the elements are examined. Combinatorial properties and the generating functions of some counting sequences associated with these…