- journal articles metadata
European Mathematical Society Publishing House
2024-03-19 03:08:04
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JCA&vol=1&iss=2&update_since=2024-03-19
Journal of Combinatorial Algebra
J. Comb. Algebra
JCA
2415-6302
2415-6310
Combinatorics
Group theory and generalizations
10.4171/JCA
http://www.ems-ph.org/doi/10.4171/JCA
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
1
2017
2
Words in linear groups, random walks, automata and P-recursiveness
Scott
Garrabrant
UCLA, LOS ANGELES, UNITED STATES
Igor
Pak
UCLA, LOS ANGELES, UNITED STATES
Cogrowth sequence of groups, probability of return, P-recursive sequences, linear groups, finite state automata
Let $S$ be a generating set of a finitely generated group $G = \langle S \rangle$. Denote by $a_n$ the number of words in $S$ of length $n$ that are equal to 1. We show that the cogrowth sequence $\{a_n\}$ is not always P-recursive. This is done by developing new combinatorial tools and using known results in computability and probability on groups.
Group theory and generalizations
Combinatorics
Probability theory and stochastic processes
Computer science
127
144
10.4171/JCA/1-2-1
http://www.ems-ph.org/doi/10.4171/JCA/1-2-1
Infinite rank spinor and oscillator representations
Steven V
Sam
University of Wisconsin, MADISON, UNITED STATES
Andrew
Snowden
University of Michigan, ANN ARBOR, UNITED STATES
Spin representations, representation stability, invariant theory
We develop a functorial theory of spinor and oscillator representations parallel to the theory of Schur functors for general linear groups. This continues our work on developing orthogonal and symplectic analogues of Schur functors. As such, there are a few main points in common. We define a category of representations of what might be thought of as the infinite rank pin and metaplectic groups, and give three models of this category in terms of: multilinear algebra, diagram categories, and twisted Lie algebras. We also define specialization functors to the finite rank groups and calculate the derived functors.
Linear and multilinear algebra; matrix theory
Combinatorics
Group theory and generalizations
145
183
10.4171/JCA/1-2-2
http://www.ems-ph.org/doi/10.4171/JCA/1-2-2
Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC
Patrick
Dehornoy
Université de Caen, CAEN, FRANCE
Artin–Titsmonoid,Artin–Tits group, gcd-monoid, enveloping group, word problem, multifraction, reduction, 3-Ore condition, type FC, embeddability, normal form, van Kampen diagram.
We describe a new approach to the word problem for Artin–Tits groups and, more generally, for the enveloping group $\mathcal U (M)$ of a monoid $M$ in which any two elements admit a greatest common divisor. The method relies on a rewrite system $\mathcal R_M$ that extends free reduction for free groups. Here we show that, if $M$ satisfies what we call the 3-Ore condition about common multiples, what corresponds to type FC in the case of Artin–Tits monoids, then the system $\mathcal R_M$ is convergent. Under this assumption, we obtain a unique representation result for the elements of $\mathcal U (M)$, extending Ore’s theorem for groups of fractions and leading to a solution of the word problem of a new type. We also show that there exist universal shapes for the van Kampen diagrams of the words representing 1.
Group theory and generalizations
Associative rings and algebras
Category theory; homological algebra
Computer science
185
228
10.4171/JCA/1-2-3
http://www.ems-ph.org/doi/10.4171/JCA/1-2-3