Related papers: Quotient singularities by permutation actions are …
We discuss here characteristic $p$ $L$-series as well as the group $S_{(q)}$ which appears to act as symmetries of these functions. We explain various actions of $S_{(q)}$ that arise naturally in the theory as well as extensions of these…
The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The main object of this paper is to obtain a characterization for all finite groups of odd order with commutativity degree…
We show that mixed-characteristic and equi-characteristic small deformations of 3-dimensional canonical (resp. terminal) singularities with perfect residue field of characteristic $p>5$ are canonical (resp. terminal). We discuss…
A finitely generated commutative monoid is uniquely presented if it has only a minimal presentation. We give necessary and sufficient conditions for finitely generated, combinatorially finite, cancellative, commutative monoids to be…
The $F$-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong $F$-regularity. However, it is very difficult to…
Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can, in…
Using the fact that any linear representation of a group can be embedded into permutations, we propose a constructive description of quantum behavior that provides, in particular, a natural explanation of the appearance of complex numbers…
Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its…
Take a closed monotone symplectic manifold containing a smooth anticanonical divisor. The quantum connection on its cohomology has singularities at zero and infinity (in the quantum parameter). At zero it has a regular singular point, by…
All unitary representations of the quantum ``az+b'' group are found. It turns out that this quantum group is self dual i.e. all unitary representations are 'numbered' by elements of the same group. Moreover, the formula for all unitary…
We study linear and hermitian representations of finite $C_2$-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an $\infty$-category. We also prove that the…
In this paper, we investigate permutation polynomials over the finite field $\mathbb F_{q^n}$ with $q=2^m$, focusing on those in the form $\mathrm{Tr}(Ax^{q+1})+L(x)$, where $A\in\mathbb F_{q^n}^*$ and $L$ is a $2$-linear polynomial over…
We show that finite dimensional half-quantum groups at roots of unity corresponding to simple Lie algebras having symmetric Cartan matrix are of wild representation type, except for sl_2. Moreover, the underlying associative algebra is…
We prove a conjecture of Koll\'ar stating that the local fundamental group of a klt singularity $x$ is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of $x$ is…
We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show…
We consider two problems that appear at first sight to be unrelated. The first problem is to count certain diagrams consisting of noncrossing arcs in the plane. The second problem concerns the weak order on the symmetric group. Each…
We give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups. We also give examples of finite quantum groups which are not…
The coinvariant algebra is a quotient of the polynomial ring $\mathbb{Q}[x_1,\ldots,x_n]$ whose algebraic properties are governed by the combinatorics of permutations of length $n$. A word $w = w_1 \dots w_n$ over the positive integers is…
As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using this discrepancy we can define a canonical singularity and a log canonical singularity. In the same way, by using a new notion, Mather-Jacobian…
The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also…