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We consider some general aspects of the new noncommutative or quantum geometry coming out of the theory of quantum groups, in connection with Planck scale physics. A generalisation of Fourier or wave-particle duality on curved spaces…
We establish a Schur-Weyl duality between a shifted quantum affine algebra and an Ariki-Koike algebra. Then, we realize a cyclotomic $q$-Schur algebra in the context of the Schur-Weyl duality.
Let $G$ be a complex linear algebraic group, $\mathfrak{g}=\Lie(G)$ its Lie algebra and $e\in\mathfrak{g}$ a nilpotent element. Vust's theorem says that in case of $G=\GL(V)$, the algebra $\mbox{End}_{G_e}(V^{\otimes d})$, where $G_e\subset…
We describe a model element able to perform universal stochastic approximations of continuous multivariable functions in both neuron-like and quantum form. The implementation of this model in the form of a multi-barrier, multiple-slit…
We study the action of the dilatation operator on restricted Schur polynomials labeled by Young diagrams with p long columns or p long rows. A new version of Schur-Weyl duality provides a powerful approach to the computation and…
The central topic of this thesis is two dimensional Quantum Gravity and its properties. The term Quantum Gravity itself is ambiguous as there are many proposals for its correct formulation and none of them have been verified experimentally.…
In this paper the authors introduce a new notion called the quantum wreath product, which is the algebra $B \wr_Q \mathcal{H}(d)$ produced from a given algebra $B$, a positive integer $d$, and a choice $Q=(R,S,\rho,\sigma)$ of parameters.…
Preparing symmetry-adapted initial states is a principal bottleneck in first-quantized quantum simulation. We present a universal approach that efficiently maps any polynomial-size superposition of occupation-number configurations to the…
The Bloch sphere representation is a geometric model for all possible quantum states of a two-level system that can be used to describe the time dynamics of a qubit. As explicit application, we consider the time dynamics of a particle in a…
Classical simulation of quantum physics is a central approach to investigating physical phenomena. Quantum computers enhance computational capabilities beyond those of classical resources, but it remains unclear to what extent existing…
We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic…
It is known that electric-magnetic duality transformations are symmetries of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. In Seiberg-Witten theory the solutions to these equations come in certain sets according to…
Fractional calculus has become an essential framework in geophysics, optics, and biological systems to capture long-range correlations and anomalous transport. In this article, we extend fractional calculus to explore a particle in a…
We prove an integral version of the Schur--Weyl duality between the specialized Birman--Murakami--Wenzl algebra $B_n(-q^{2m+1},q)$ and the quantum algebra associated to the symplectic Lie algebra sp_{2m}. In particular, we deduce that this…
Unitary Coined Discrete-Time Quantum Walks (UC-DTQW) constitute a universal model of quantum computation, meaning that any computation done by a general purpose quantum computer can either be done using the UC-DTQW framework. In the last…
The classical Mycielski transformation allows for constructing from a given graph the new one, with an arbitrarily large chromatic number but preserving the size of the largest clique contained in it. This particular construction and its…
There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation…
We show that the Schur-Weyl type duality between $gl(1|1)$ and $GL_n$ gives a natural representation-theoretic setting for the relation between reduced and non-reduced Burau representations.
Quantum mechanics has led not only to new physical theories, but also a new understanding of information and computation. Quantum information began by yielding new methods for achieving classical tasks such as factoring and key distribution…
Quantum graphs are a paradigmatic model for quantum chaos as well as for spectral theory. We give a concise didactical introduction to quantum graphs, or Schr\"odinger Hamiltonians on metric graphs, with a focus on results related to…