Related papers: On the Mathieu Conjecture for $SU(N)$ and $SO(N)$

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes. In the 1980's Thompson conjectured that the equality $N(G)=N(S)$, where $Z(G)=1$ and $S$ is simple, implies the isomorphism $G\simeq S$. In a series of papers of…

For $n \ge 2$, Gamburd, Jakobson, and Sarnak [J. Eur. Math. Soc. 1, 51-85 (1999)] conjectured that almost every $n$-tuple in $\mathrm{SU}(2)$ has a spectral gap. Toward this conjecture, Fisher [Int. Math. Res. Not. (2006)] established a…

In 1987, Mazur and Tate stated conjectures which, in some cases, resemble the classical Birch-Swinnerton-Dyer conjecture and its $p$-adic analog. We study experimentally three conjectures stated by Mazur and Tate using SageMath. Our…

Let S be a smooth projective surface equipped with a line bundle H. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to H on the Hilbert scheme of points of S. Voisin has recently reduced Lehn's…

We introduce some generalized topological concepts to deal with union-closed families, and show that one can reduce the proof of Frankl's conjecture to some families of so-called supratopological spaces. We prove some results on the…

Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is the unique holomorphic normalized…

We show that the correspondence between SU(5) monopoles and the elementary particles, which underlies the construction of a dual standard model, has some simpler analogues associated with the strong, weak and hypercharge interactions. We…

We examine the low energy structure of N=1 supersymmetric SO(10) gauge theory with matter chiral superfields in N_Q spinor and N_f vector representations. We construct a dual to this model based upon an SU(N_f+2N_Q-7) x Sp(2N_Q-2) gauge…

We investigate the duality between the Nekrasov function and the quantized Seiberg-Witten prepotential, first guessed in [1] and further elaborated in [2] and [3]. We concentrate on providing more thorough checks than the ones presented in…

A generalized Euler parameterization of a compact Lie group is a way for parameterizing the group starting from a maximal Lie subgroup, which allows a simple characterization of the range of parameters. In the present paper we consider the…

We introduce (generalised) Pfaffian operators into our lattice calculations of the mass spectra and confining string tensions of SO(2N) gauge theories, complementing the conventional trace operators used in previous lattice calculations. In…

Bouc proposed the following conjecture: a finite group $G$ is nilpotent if and only if its largest quotient $B$-group $\beta(G)$ is nilpotent. And he has prove that this conjecture holds when $G$ is solvable. In this paper, we consider the…

Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrodinger's equation. Mathieu's equation can be easily solved…

In seven dimensions any spin manifold admits an SU(2) structure and therefore very general M-theory compactifications have the potential to allow for a reduction to N=4 gauged supergravity. We perform this general SU(2) reduction and give…

We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning Euler systems for the multiplicative group over arbitrary number fields.

An elementary recursive relation for M$\ddot{\mathrm{o}}$bius function $\mu (n)$ is introduced by two simple ways. With this recursive relation, $\mu (n)$ can be calculated without directly knowing the factorization of the $n$. $\mu (1)…

Let p be an odd prime and S a finite p-group. B. Oliver's conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup X(S) of S always contains the Thompson subgroup.…

We review and extend the results of [1] that gives a condition for reducibility of quantum representations of mapping class groups constructed from Reshetikhin-Turaev type topological quantum field theories based on modular categories. This…

The descent method is one of the approaches to study the Brauer--Manin obstruction to the local--global principle and to weak approximation on varieties over number fields, by reducing the problem to ``descent varieties''. In recent lecture…

We study the possibility of family unification on the basis of SO(2N) gauge theory on the five-dimensional space-time, $M^4\times S^1/Z_2$. Several SO(10), $SU(4) \times SU(2)_L \times SU(2)_R$ or SU(5) multiplets come from a single bulk…