Related papers: On minimal bases in homotopical combinatorics

We provide a new combinatorial approach to studying the collection of N-infinity-operads in G-equivariant homotopy theory for G a finite cyclic group. In particular, we show that for G the cyclic group of order p^n the natural order on the…

We introduce an explicit combinatorial characterization of the minimal model ${\cal O}_{\infty}$ of the coloured operad ${\cal O}$ encoding non-symmetric operads. In our description of ${\cal O}_{\infty}$, the spaces of operations are…

We provide a general recursive method for constructing transfer systems on finite lattices. Using this we calculate the number of homotopically distinct $N_\infty$ operads for dihedral groups $D_{p^n}$, $p > 2$ prime, and cyclic groups…

We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study…

This chapter provides a comprehensive survey of foundational results and recent advances concerning minimal generating sets for the mapping class group of a nonorientable surface, $\mathrm{Mod}(N_{g})$, and its index-two twist subgroup,…

We prove the existence of minimal models a la Sullivan for operads with non-trivial arity zero. So, up to homotopy algebras with strict units are just operad algebras over these minimal models. As an application we give another proof of the…

We prove a new case of Rubin's saturation conjecture about the realization of $G$-transfer systems, for $G$ a finite cyclic group, by linear isometries $N_\infty$-operads, namely the case of cyclic groups of order $p^nq^m$ for $p,q$…

This paper is a survey on the homotopy theory of $E_n$-operads written for the new handbook of homotopy theory.

When we consider a finite abelian group acting linearly on a polynomial ring, we can find monomial generators for the subring of invariants. By Noether's degree bound and Hilbert's finiteness theorem, we know that there are finitely many…

Over a field of characteristic $0$, we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of $n \times n$ matrices. This modifies a conjecture of Weyman and provides a complete…

We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups $G$ to rank one subgroups $G_1$. For this we use the realizations of complementary…

We investigate closure operators and describe their properties for $E$-combinations and $P$-combinations of structures and their theories. We prove, for $E$-combinations, that the existence of a minimal generating set of theories is…

It is a classical problem to compute a minimal set of invariant polynomial generating the invariant ring of a finite group as an algebra. We present here an algorithm for the computation of minimal generating sets in the non-modular case.…

We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of…

Let $\mathcal{G}$ be a countably infinite group of unitary operators on a complex separable Hilbert space $H$. Let $X = \{x_{1},...,x_{r}\}$ and $Y = \{y_{1},...,y_{s}\}$ be finite subsets of $H$, $r < s$, $V_{0} = \bar{span}…

For a finite group $G$, $N_\infty$ operads encode collections of norm maps, and by work of Blumberg--Hill and Rubin their homotopy category is equivalent to the poset of $G$--transfer systems on the subgroup lattice of $G$. In \cite{ABB+25}…

We show that a certain class of categorical operads give rise to $E_n$-operads after geometric realization. The main arguments are purely combinatorial and avoid the technical topological assumptions otherwise found in the literature.

We prove the existence of minimal models a la Sullivan for operads with nontrivial arity zero. So up-to-homotopy algebras with strict units are just operad algebras over these minimal models. As an application, we give another proof of the…

We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain…

Although degree bounds and algorithms for the generators of various invariant rings have been known for decades, little is known about the cardinality of minimal generating sets. Estimates of such would provide lower bounds for the runtime…