Symmetries of double ratios and an equation for Möbius structures

Orthogonal representations η n : S n ↷ R N \eta _n\colon S_n\curvearrowright \mathbb {R}^N of the symmetric groups S n S_n , n ≥ 4 n\ge 4 , with N = n ! / 8 N=n!/8 , emerging from symmetries of double ratios are treated. For n = 5 n=5 , the representation η 5 \eta _5 is decomposed into irreducible components and it is shown that a certain component yields a solution of the equations that describe the Möbius structures in the class of sub-Möbius structures. In this sense, a condition determining the Möbius structures is implicit already in symmetries of double ratios.

Optimal lobbying pricing

We study a game with two candidates and two interest groups. The groups offer two kinds of costly contributions to achieve political influence: (a) pre-election campaign contributions to their favourite candidates that increase their probability of winning the election and (b) post-election lobbying contributions to the winning candidate to affect the implemented policy. The candidates are the first to act by strategically choosing the lobbying prices they will charge the groups if they are elected. We characterise the equilibrium values of the lobbying prices set by the candidates as well as the equilibrium levels of the campaign and lobbying contributions chosen by the groups. We show, endogenously, that in the case with symmetric groups and symmetric politicians, a candidate announces to charge the group that supports her in the election a lower lobbying price, justifying this way the preferential treatment to certain groups from the politicians in office. We also consider two extensions (asymmetric groups and politicians who do not commit to the announced prices) and show that the results of the benchmark model hold under specific conditions.

Advancing mathematics by guiding human intuition with AI

The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer conjecture2, a Millennium Prize Problem3. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups4. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.

A Constructive Method for Generating Short Presentations for the Symmetric Groups Sm+n, S2m and Smn

A long-standing problem is how to create a short-length presentation for finite groups of degree n. This paper aimed at presenting a concrete method for generating presentations for the groups Sm+n, S2m and Smn for all m,nÎZ+ with fewer relations than the existing literature from the presentations of Sm and Sn. The aim is achieved by considering finite groups acting on sets and Cartesian product of groups which lead to the construction of multiple transformations as representatives of some finite groups.

Differential operators and Higher Specht polynomials

In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

Abelian sections of the symmetric groups with respect to their index

We show the existence of an absolute constant $$\alpha >0$$ α > 0 such that, for every $$k \ge 3$$ k ≥ 3 , $$G:= \mathop {\mathrm {Sym}}(k)$$ G : = Sym ( k ) , and for every $$H \leqslant G$$ H ⩽ G of index at least 3, one has $$|H/H'| \le |G:H|^{\alpha / \log \log |G:H|}$$ | H / H ′ | ≤ | G : H | α / log log | G : H | . This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.

Cochain level May–Steenrod operations

Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-i products; a family of coherent homotopies derived from the broken symmetry of Alexander–Whitney’s chain approximation to the diagonal. He later defined his homonymous operations for all primes using the homology of symmetric groups. This approach enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at the even prime. In recent years, thanks to the development of new applications of cohomology, having definitions of Steenrod operations that can be effectively computed in specific examples has become a key issue. Using the operadic viewpoint of May, this article provides such definitions at all primes introducing multioperations that generalize the Steenrod cup-i products on the simplicial and cubical cochains of spaces.