Related papers: The one-sided cycle shuffles in the symmetric grou…
The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices (n points). In this paper, we generalize many aspects of this situation. We introduce random shifts of…
We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the…
We present various results on multiplying cycles in the symmetric group. Our first result is a generalisation of the following theorem of Boccara (1980): the number of ways of writing an odd permutation in the symmetric group on $n$ symbols…
This is an introduction to the group algebras of the symmetric groups, written for a quarter-long graduate course. After recalling the definition of group algebras (and monoid algebras) in general, as well as basic properties of…
We give shuffle algebra realization of positive part of quantum affine superalgebra $U_{v}(\widehat{\mathfrak{D}}(2,1;\theta))$ associated to any simple root systems. We also determine the shuffle algebra associated to…
We use the fact that certain cosets of the stabilizer of points are pairwise conjugate in a symmetric group $S_n$ in order to construct recurrence relations for enumerating certain subsets of $S_n$. Occasionally one can find `closed form'…
It is pointed out that an exactly solvable permutation operator, viewed as the quantization of cyclic shifts, is useful in constructing a basis in which to study the quantum baker's map, a paradigm system of quantum chaos. In the basis of…
Given the symmetric group $G = \operatorname{Sym}(n)$ and a multiplicity-free subgroup $H\leq G$, the orbitals of the action of $G$ on $G/H$ by left multiplication induce a commutative association scheme. The irreducible constituents of the…
The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order,…
Shuffle algebras are monoids for an unconvential monoidal category structure on graded vector spaces. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on…
Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained.…
We construct a family of PBWD bases for the positive subalgebras of quantum loop algebras of type $B_n$ and $G_2$, as well as their Lusztig and RTT (for type $B_n$ only) integral forms, in the new Drinfeld realization. We also establish a…
Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we aim to generalise these results…
Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network…
In this paper we will give a similar factorization as in \cite{4}, \cite{5}, where the autors Svrtan and Meljanac examined certain matrix factorizations on Fock-like representation of a multiparametric quon algebra on the free associative…
We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al., and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial…
We describe an efficient algorithm to write any element of the alternating group A_n as a product of two n-cycles (in particular, we show that any element of A_n can be so written -- a result of E. A. Bertram). An easy corollary is that…
We define a quantum loop group $\mathbf{U}^+_Q$ associated to an arbitrary quiver $Q=(I,E)$ and maximal set of deformation parameters, with generators indexed by $I \times \mathbb{Z}$ and some explicit quadratic and cubic relations. We…
The shuffle algebra on positive integers encodes the usual multiple zeta values (MZVs) (with positive arguments) thanks to the representations of MZVs by iterated Chen integrals of Kontsevich. Together with the quasi-shuffle (stuffle)…
Many seemingly disparate Markov chains are unified when viewed as random walks on the set of chambers of a hyperplane arrangement. These include the Tsetlin library of theoretical computer science and various shuffling schemes. If only…