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In this paper, we initiate a study into the explicit construction of irreducible representations of the Hecke algebra $H_n(q)$ of type $A_{n-1}$ in the non-generic case where $q$ is a root of unity. The approach is via the Specht modules of…

For a subgroup of $PGL(2,q)$ we show how some irreducible polynomials over $\mathbb{F}_q$ arise from the field of invariant rational functions. The proofs rely on two actions of $PGL(2,F)$, one on the projective line over a field $F$ and…

In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,x_2,\cdots,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary…

The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…

We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the…

In this paper, we present a linear algebraic approach to the study of permutation polynomials that arise from linear maps over a finite field $\mathbb{F}_{q^2}$. We study a particular class of permutation polynomials over…

The Eigendecomposition of quadratic forms (symmetric matrices) guaranteed by the spectral theorem is a foundational result in applied mathematics. Motivated by a shared structure found in inferential problems of recent interest---namely…

Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x_0 be a sufficiently general element of K, and let alpha be a root of f. We give precise conditions under which Newton iteration, started at the point…

For each prime p other than 3, and each power q=p^k, we present two large classes of permutation polynomials over F_{q^2} of the form X^r B(X^{q-1}) which have at most five terms, where B(X) is a polynomial with coefficients in {1,-1}. The…

Let $q$ be a prime power and $\mathbb F_{q^n}$ be the finite field with $q^n$ elements, where $n>1$. We introduce the class of the linearized polynomials $L(x)$ over $\mathbb F_{q^n}$ such that…

We develop a tree method for multidimensional q-Hahn polynomials. We define them as eigenfunctions of a multidimensional q-difference operator and we use the factorization of this operator as a key tool. Then we define multidimensional…

The use of permutation polynomials has appeared, along to their compositional inverses, as a good choice in the implementation of cryptographic systems. Hence, there has been a demand for constructions of these polynomials which…

Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical…

From analysis of a big variety of different knots we conclude that at q which is an root of unity, q^{2m}=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: H_{r+m} = H_r H_m for any A, which is a…

An algorithm for factoring polynomials over finite fields is given by Berlekamp in 1967. The main tool was the matrix Q corresponding to each polynomial. This paper studies the degrees of polynomials over binary field that associated with…

This paper shows that orbital equations generated by iteration of polynomial maps do not have necessarily a unique representation. Remarkably, they may be represented in an infinity of ways, all interconnected by certain nonlinear…

We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms.…

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of…

We study the decomposition of tensor powers of two dimensional irreducible representations of quantum $\mathfrak{sl}_2$ at even roots of unity into direct sums of tilting modules. We derive a combinatorial formula for multiplicity of…

Let $Q$ be a quiver of $A_n$ type and $\mathbb{K}$ be an algebraically closed field. A nilpotent endomorphism of a quiver representation induces a linear transformation of the vector space at each vertex. Generically among all nilpotent…