Universal rings arising in geometry and group theory

In this paper homotopical methods for the description of subgroups determined by ideals in group rings are introduced. It is shown that in certain cases the subgroups determined by symmetric product of ideals in group rings can be described with the help of homotopy groups of spheres.

Let $K$ be an algebraically closed field of characteristic zero. Algebraic structures of a specific type (e.g. algebras or coalgebras) on a given vector space $W$ over $K$ can be encoded as points in an affine space $U(W)$. This space is equipped with a $\text{GL}(W)$ action, and two points define isomorphic structures if and only if they lie in the same orbit. This leads to study the ring of invariants $K[U(W)]^{\text{GL}(W)}$. We describe this ring by generators and relatio...

This paper studies the integral cohomology ring of the classifying space $BPU_n$ of the projective unitary group $PU_n$. By calculating a Serre spectral sequence, we determine the ring stucture of $H^*(BPU_n;\mathbb{Z})$ in dimensions $\leq 11$. For any odd prime $p$, we also determine the $p$-primary subgroups of $H^i(BPU_n;\mathbb{Z})$ in the range $i\leq 2p+13$ for $i$ odd and $i\leq 4p+8$ for $i$ even. The main technique used in the calculation is applying the theory of Y...

This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric f...

The Jucys-Murphy elements for wreath products G_n associated to any finite group G are introduced and they play an important role in our study on the connections between class algebras of G_n for all n and vertex algebras. We construct an action of (a variant of) the W_{1+\infty} algebra acting irreducibly on the direct sum R_G of the class algebras of G_n for all n in a group theoretic manner. We establish various relations between convolution operators using JM elements and...

Let $R$ be a commutative ring of characteristic zero and $G$ an arbitrary group. In the present paper we classify the groups $G$ for which the set of symmetric elements with respect to the classical involution of the group ring $RG$ is Lie metabelian.

This is a written version of the invited lecture at the 9th European Congress of Mathematics in July 2024 in Sevilla. We review certain new symmetries of Grothendieck rings that have emerged in representation theory.

Let H_c be the rational Cherednik algebra of type A_{n-1} with spherical subalgebra U_c = eH_ce. Then U_c is filtered by order of differential operators, with associated graded ring gr U_c = C[h+h*]^W, where W is the n-th symmetric group. We construct a filtered Z-algebra B such that, under mild conditions on c: (1) The category B-qgr of graded noetherian B-modules modulo torsion is equivalent to U_c-mod; (2) The associated graded Z-algebra gr(B) has gr(B)-qgr equivalent ...

In this paper we introduce the semi-graded rings, which extend graded rings and skew PBW extensions. For this new type of non-commutative rings we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand-Kirillov dimension. We will extended the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the...

Macaulay posets are posets in which an analog of the Kruskal-Katona Theorem holds. Macaulay rings (also called Macaulay-Lex rings) are rings in which an analog of Macaulay's Theorem for lex ideals holds. The study of both of these objects started with Macaulay almost a century ago. Since then, these two branches have developed separately over the past century, with the last link being the Clements-Lindstr\"om Theorem. For every ring that is the quotient of a polynomial ring...