- journal articles metadata
European Mathematical Society Publishing House
2024-03-19 13:22:59
11
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JCA&vol=1&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
1
Some unitary representations of Thompson’s groups $F$ and $T$
Vaughan
Jones
Vanderbilt University, NASHVILLE, UNITED STATES
Thompson group, subfactor, conformal field theory, diffeomorphism, planar algebra, partition function, knot, link, Seifert surface
In a “naive” attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson’s groups $T$ and $F$ for any subfactor. The Thompson group elements are the “local scale transformations” of the theory. In a simple case the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new “oriented” subgroups of $\overrightarrow F < F$ and $\overrightarrow T < T$ which allow us to produce all oriented knots and links.
Group theory and generalizations
Functional analysis
Manifolds and cell complexes
Quantum theory
1
44
10.4171/JCA/1-1-1
http://www.ems-ph.org/doi/10.4171/JCA/1-1-1
The canonical basis of the quantum adjoint representation
George
Lusztig
MIT, CAMBRIDGE, UNITED STATES
Quantum group, canonical basis, Chevalley group
We identify the canonical basis of the quantum adjoint representation of a quantized enveloping algebra with a basis that we defined before the theory of canonical bases was available.
Group theory and generalizations
Topological groups, Lie groups
45
57
10.4171/JCA/1-1-2
http://www.ems-ph.org/doi/10.4171/JCA/1-1-2
Kazhdan groups whose FC-radical is not virtually abelian
Mikhail
Ershov
University of Virginia, CHARLOTTESVILLE, UNITED STATES
Kazhdan's property (T), Golod–Shafarevich groups, FC-radical
We construct examples of residually finite groups with Kazhdan's property ($T$) whose FC-radical is not virtually abelian. This answer a question of Popa and Vaes about possible fundamental groups of II$_1$ factors arising from Kazhdan groups.
Group theory and generalizations
59
62
10.4171/JCA/1-1-3
http://www.ems-ph.org/doi/10.4171/JCA/1-1-3
Quantum Satake in type $A$. Part I
Ben
Elias
University of Oregon, EUGENE, UNITED STATES
Geometric Satake, quantum, Soergel bimodules, diagrammatics, webs
We give an interpretation of $\mathfrak {sl}_n$-webs as morphisms between certain singular Soergel bimodules. We explain how this is a combinatorial, algebraic version of the geometric Satake equivalence (in type $A$). We then $q$-deform the construction, giving an equivalence between representations of $U_q (\mathfrak {sl}_n)$ and certain singular Soergel bimodules for a $q$-deformed Cartan matrix. In this paper, we discuss the general case but prove only the case $n = 2, 3$. In the sequel we will prove $n \geq 4$.
Nonassociative rings and algebras
Group theory and generalizations
63
125
10.4171/JCA/1-1-4
http://www.ems-ph.org/doi/10.4171/JCA/1-1-4
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
3
Multifraction reduction II: Conjectures for Artin–Tits groups
Patrick
Dehornoy
Université de Caen and and Institut Universitaire de France, France
Artin–Tits monoid, Artin–Tits group, gcd-monoid, enveloping group, word problem, multifraction, reduction, semi-convergence, cross-confluence, tame reduction, van Kampen diagram, embeddability
Multifraction reduction is a new approach to the word problem for Artin–Tits groups and, more generally, for the enveloping group of a monoid in which any two elements admit a greatest common divisor. This approach is based on a rewrite system (“reduction”) that extends free group reduction. In this paper, we show that assuming that reduction satisfies a weak form of convergence called semi-convergence is sufficient for solving the word problem for the enveloping group, and we connect semi-convergence with other conditions involving reduction. We conjecture that these properties are valid for all Artin–Tits monoids, and provide partial results and numerical evidence supporting such conjectures.
Group theory and generalizations
Category theory; homological algebra
Computer science
229
287
10.4171/JCA/1-3-1
http://www.ems-ph.org/doi/10.4171/JCA/1-3-1
Lie algebras and torsion groups with identity
Efim
Zelmanov
University of California – San Diego, La Jolla, USA
The Burnside problem, pro-$p$ groups, PI-algebras, Lie algebras
We prove that a finitely generated Lie algebra $L$ such that (i) every commutator in generators is ad-nilpotent, and (ii) $L$ satisfies a polynomial identity, is nilpotent. As a corollary we get that a finitely generated residually-$p$ torsion group whose pro-$p$ completion satisfies a pro-$p$ identity is finite.
Group theory and generalizations
Associative rings and algebras
289
340
10.4171/JCA/1-3-2
http://www.ems-ph.org/doi/10.4171/JCA/1-3-2
4
Multifraction reduction III: The case of interval monoids
Patrick
Dehornoy
Université de Caen, France
Friedrich
Wehrung
Université de Caen, France
Poset, intervalmonoid, gcd-monoid, enveloping group, word problem,multifraction, reduction, embeddability, semi-convergence, circuit, zigzag
We investigate gcd-monoids, which are cancellative monoids in which any two elements admit a left and a right gcd, and the associated reduction of multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the word problemfor the enveloping group. Here we consider the particular case of interval monoids associated with finite posets. In this way, we construct gcd-monoids, in which reduction of multifractions has prescribed properties not yet known to be compatible: semi-convergence of reductionwithout convergence, semi-convergence up to some level but not beyond, non-embeddability into the enveloping group (a strong negation of semi-convergence).
Order, lattices, ordered algebraic structures
Category theory; homological algebra
Group theory and generalizations
Computer science
341
370
10.4171/JCA/1-4-1
http://www.ems-ph.org/doi/10.4171/JCA/1-4-1
Young tableaux and representations of Hecke algebras of type ADE
Loïc
Poulain d'Andecy
Université de Reims Champagne-Ardenne, France
Hecke algebras, Weyl groups, simply-laced root systems, affine Hecke algebras, Jucys–Murphy elements, skew partitions, Young tableaux, seminormal representations, calibrated representations
We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of type ADE. Representations are constructed by explicit formulas, in a seminormal form.
Group theory and generalizations
Combinatorics
371
423
10.4171/JCA/1-4-2
http://www.ems-ph.org/doi/10.4171/JCA/1-4-2