arXiv subjects

Fatemeh Bagherzadeh

Publications and source records attributed to Fatemeh Bagherzadeh.

6 recordsLinked to original sources

arXiv2025

Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube

We study a two-parameter generalization of the Catalan numbers: $C_{d,p}(n)$ is the number of ways to subdivide the $d$-dimensional hypercube into $n$ rectangular blocks using orthogonal partitions of fixed arity $p$. Bremner \& Dotsenko introduced $C_{d,p}(n)$ in their work on Boardman--Vogt tensor products of operads; they used homological algebra to prove a recursive formula and a functional equation. We express $C_{d,p}(n)$ as simple finite sums, and determine their growth rate and asymptotic behaviour. We give an elementary proof of the functional equation, using a bijection between hypercube decompositions and a family of full $p$-ary trees. Our results generalize the well-known correspondence between Catalan numbers and full binary trees.

math.CO↗

arXiv2018

Operadic approach to cohomology of associative triple and N-tuple systems

The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n = 3 and n = 4, we provide an explicit definition of this cup product and prove its basis properties.

math.CT↗

arXiv2018

Gröbner bases and dimension formulas for ternary partially associative operads

Dotsenko and Vallette discovered an extension to nonsymmetric operads of Buchberger's algorithm for Gröbner bases of polynomial ideals. In the free nonsymmetric operad with one ternary operation $({\ast}{\ast}{\ast})$, we compute a Gröbner basis for the ideal generated by partial associativity $((abc)de) + (a(bcd)e) + (ab(cde)$. In the category of $\mathbb{Z}$-graded vector spaces with Koszul signs, the (homological) degree of $({\ast}{\ast}{\ast})$ may be even or odd. We use the Gröbner bases to calculate the dimension formulas for these operads.

math.RA↗

arXiv2017

Classification of finite W-groups

We determine the structure of the W-group $\mathcal{G}_F$, the small Galois quotient of the absolute Galois group $G_F$ of the Pythagorean formally real field $F$ when the space of orderings $X_F$ has finite order. Based on Marshall's work (1979), we reduce the structure of $\mathcal{G}_F$ to that of $\mathcal{G}_{\bar{F}}$, the W-group of the residue field $\bar{F}$ when $X_F$ is a connected space. In the disconnected case, the structure of $\mathcal{G}_F$ is the free product of the W-groups $\mathcal{G}_{F_i}$ corresponding to the connected components $X_i$ of $X_F$. We also give a completely Galois theoretic proof for Marshall's Basic Lemma.

math.NT↗

arXiv2017

Commutativity in double interchange semigroups

We extend the work of Kock (2007) and Bremner & Madariaga (2016) on commutativity in double interchange semigroups (DIS) to relations with 10 arguments. Our methods involve the free symmetric operad generated by two binary operations with no symmetry, its quotient by the two associative laws, its quotient by the interchange law, and its quotient by all three laws. We also consider the geometric realization of free double interchange magmas by rectangular partitions of the unit square $I^2$. We define morphisms between these operads which allow us to represent elements of free DIS both algebraically as tree monomials and geometrically as rectangular partitions. With these morphisms we reason diagrammatically about free DIS and prove our new commutativity relations.

math.RA↗

arXiv2016

Jordan Trialgebras and Post-Jordan Algebras

We compute minimal sets of generators for the S_n-modules (n <= 4) of multilinear polynomial identities of arity n satisfied by the Jordan product and the Jordan diproduct (resp. pre-Jordan product) in every triassociative (resp. tridendriform) algebra. These identities define Jordan trialgebras and post-Jordan algebras: Jordan analogues of the Lie trialgebras and post-Lie algebras introduced by Dotsenko et al., Pei et al., Vallette & Loday. We include an extensive review of analogous structures existing in the literature, and their interrelations, in order to identify the gaps filled by our two new varieties of algebras. We use computer algebra (linear algebra over finite fields, representation theory of symmetric groups), to verify in both cases that every polynomial identity of arity <= 6 is a consequence of those of arity <= 4. We conjecture that in both cases the next independent identities have arity 8, imitating the Glennie identities for Jordan algebras. We formulate our results as a commutative square of operad morphisms, which leads to the conjecture that the squares in a much more general class are also commutative.

math.RA↗