More Like this

1. Hopf Monoids and Generalized Permutahedra

   https://doi.org/10.1090/memo/1437  

   Aguiar, Marcelo; Ardila, Federico (September 2023, Memoirs of the American Mathematical Society)

   Generalized permutahedra are polytopes that arise in combinatorics, algebraic geometry, representation theory, topology, and optimization. They possess a rich combinatorial structure. Out of this structure we build a Hopf monoid in the category of species. Species provide a unifying framework for organizing families of combinatorial objects. Many species carry a Hopf monoid structure and are related to generalized permutahedra by means of morphisms of Hopf monoids. This includes the species of graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, and building sets, among others. We employ this algebraic structure to define and study polynomial invariants of the various combinatorial structures. We pay special attention to the antipode of each Hopf monoid. This map is central to the structure of a Hopf monoid, and it interacts well with its characters and polynomial invariants. It also carries information on the values of the invariants on negative integers. For our Hopf monoid of generalized permutahedra, we show that the antipode maps each polytope to the alternating sum of its faces. This fact has numerous combinatorial consequences. We highlight some main applications: We obtain uniform proofs of numerous old and new results about the Hopf algebraic and combinatorial structures of these families. In particular, we give optimal formulas for the antipode of graphs, posets, matroids, hypergraphs, and building sets. They are optimal in the sense that they provide explicit descriptions for the integers entering in the expansion of the antipode, after all coefficients have been collected and all cancellations have been taken into account. We show that reciprocity theorems of Stanley and Billera–Jia–Reiner (BJR) on chromatic polynomials of graphs, order polynomials of posets, and BJR-polynomials of matroids are instances of one such result for generalized permutahedra. We explain why the formulas for the multiplicative and compositional inverses of power series are governed by the face structure of permutahedra and associahedra, respectively, providing an answer to a question of Loday. We answer a question of Humpert and Martin on certain invariants of graphs and another of Rota on a certain class of submodular functions. We hope our work serves as a quick introduction to the theory of Hopf monoids in species, particularly to the reader interested in combinatorial applications. It may be supplemented with Marcelo Aguiar and Swapneel Mahajan’s 2010 and 2013 works, which provide longer accounts with a more algebraic focus. 

   more » « less
2. Polynomial Representations of the Witt Lie Algebra

   https://doi.org/10.1093/imrn/rnae139  

   Sam, Steven_V; Snowden, Andrew; Tosteson, Philip (June 2024, International Mathematics Research Notices)

   Abstract The Witt algebra  $${\mathfrak{W}}_{n}$$ is the Lie algebra of all derivations of the $$n$$-variable polynomial ring $$\textbf{V}_{n}=\textbf{C}[x_{1}, \ldots , x_{n}]$$ (or of algebraic vector fields on $$\textbf{A}^{n}$$). A representation of $${\mathfrak{W}}_{n}$$ is polynomial if it arises as a subquotient of a sum of tensor powers of $$\textbf{V}_{n}$$. Our main theorems assert that finitely generated polynomial representations of $${\mathfrak{W}}_{n}$$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $$\textbf{Fin}^{\textrm{op}}$$, where $$\textbf{Fin}$$ is the category of finite sets. We also show that polynomial representations of $${\mathfrak{W}}_{n}$$ are equivalent to polynomial representations of the endomorphism monoid of $$\textbf{A}^{n}$$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish. 

   more » « less
3. A weaker notion of the finite factorization property

   Jiang, Henry; Kanungo, Shihan; Kim, Harry (April 2024, Communications of the Korean Mathematical Society)

   An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid M is a length-finite factorization monoid if each b ∈ M has only finitely many factorizations of any prescribed length. An additive submonoid of ℝ≥0 is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years. 

   more » « less
4. On the atomic structure of torsion-free monoids

   https://doi.org/10.1007/s00233-023-10385-8  

   Gotti, Felix; Vulakh, Joseph (October 2023, Semigroup Forum)

   Abstract LetMbe a cancellative and commutative (additive) monoid. The monoidMis atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also,Msatisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids. 

   more » « less
5. Algebraic Systems Motivated by DNA Origami

   Garrett, J.; Jonoska, N.; Saito, M (January 2019, Algebraic Informatics)

   Ćirić, M.; Droste, M.; Pin, JÉ. (Ed.)

   We initiate an algebraic approach to study DNA origami structures. We identify two types of basic building blocks and describe a DNA origami structure by their composition. These building blocks are taken as generators of a monoid, called the origami monoid, and motivated by the well studied Temperley-Lieb algebras, we identify a set of relations that characterize the origami monoid. We present several observations about Green’s relations for the origami monoid and study the relations to a direct product of Jones monoids, which is a morphic image of an origami monoid. 

   more » « less