- journal articles metadata
European Mathematical Society Publishing House
2024-03-19 12:15:47
13
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JCA&vol=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
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
2
2018
1
On joins and intersections of subgroups in free groups
Sergei
Ivanov
University of Illinois, Urbana, USA
Free groups, Stallings graphs, intersections and joins of subgroups
We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and Müller on these graphs, how to repair the lemma and how to utilize it.
Group theory and generalizations
Manifolds and cell complexes
1
18
10.4171/JCA/2-1-1
http://www.ems-ph.org/doi/10.4171/JCA/2-1-1
2
8
2018
The periplectic Brauer algebra II: Decomposition multiplicities
Kevin
Coulembier
University of Sydney, Australia
Michael
Ehrig
University of Sydney, Australia
Periplectic Lie superalgebra, periplectic Brauer algebra, decomposition multiplicities, (skew) Young diagrams, standardly based algebras
We determine the Jordan–Hölder decomposition multiplicities of projective and cell modules over periplectic Brauer algebras in characteristic zero. These are obtained by developing the combinatorics of certain skew Young diagrams.We also establish a useful relationship with the Kazhdan–Lusztig multiplicities of the periplectic Lie supergroup.
Associative rings and algebras
Nonassociative rings and algebras
Quantum theory
19
46
10.4171/JCA/2-1-2
http://www.ems-ph.org/doi/10.4171/JCA/2-1-2
2
8
2018
On skew braces (with an appendix by N. Byott and L. Vendramin)
Agata
Smoktunowicz
University of Edinburgh, UK
Leandro
Vendramin
University of Buenos Aires, Argentina
Braces, Yang–Baxter, rings, near-rings, triply factorized groups, matched pair of groups, bijective 1-cocycles, Hopf–Galois extensions
Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang–Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf–Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.
Associative rings and algebras
Quantum theory
47
86
10.4171/JCA/2-1-3
http://www.ems-ph.org/doi/10.4171/JCA/2-1-3
2
8
2018
Remarks on profinite groups having few open subgroups
Dan
Segal
Oxford University, UK
Profinite groups, strong completeness
We explore the relations between various conditions of „smallness" on profinite groups.
Group theory and generalizations
87
101
10.4171/JCA/2-1-4
http://www.ems-ph.org/doi/10.4171/JCA/2-1-4
2
8
2018
2
The $\mathfrak{sl}_\infty$-crystal combinatorics of higher level Fock spaces
Thomas
Gerber
RWTH Aachen, Germany
Emily
Norton
Max Planck Institute for Mathematics, Bonn, Germany
Fock space, crystals, representations of rational Cherednik algebras, Heisenberg algebra, combinatorial classification of supports
For integers $e,\ell\geq 2$, the level $\ell$ Fock space has an $\mathfrak{sl}_\infty$-crystal structure arising from the action of a Heisenberg algebra, intertwining the $\widehat{\mathfrak{sl}_e}$-crystal. The vertices of these crystals are charged $\ell$-partitions. We give the combinatorial rule for computing the arrows anywhere in the $\mathfrak{sl}_\infty$-crystal. This allows us to pinpoint the location of any charged $\ell$-partition. As an application, we compute the support of the spherical representation of a cyclotomic rational Cherednik algebra, and in particular, the set of parameters such that it is finite-dimensional. We also give an easy abacus characterization of all finite-dimensional representations of type $B$ Cherednik algebras.
Combinatorics
Nonassociative rings and algebras
Group theory and generalizations
103
145
10.4171/JCA/2-2-1
http://www.ems-ph.org/doi/10.4171/JCA/2-2-1
5
8
2018
Quadratic automaton algebras and intermediate growth
Natalia
Iyudu
Universty of Edinburgh, UK
Stanislav
Shkarin
Queen's University Belfast, UK
Automaton algebras, word combinatorics, regular language, Gröbner basis, quadratic algebras, Koszul algebras, finitely presented algebras, Hilbert series, Gelfand–Kirillov dimension
We present an example of a quadratic algebra given by three generators and three relations, which is automaton (the set of normal words forms a regular language) and such that its ideal of relations does not possess a finite Gröbner basis with respect to any choice of generators and any choice of a well-ordering of monomials compatible withmultiplication. This answers a question of Ufnarovski. Another result is a simple example (4 generators and 7 relations) of a quadratic algebra of intermediate growth.
Associative rings and algebras
Nonassociative rings and algebras
147
167
10.4171/JCA/2-2-2
http://www.ems-ph.org/doi/10.4171/JCA/2-2-2
5
8
2018
A DG-extension of symmetric functions arising from higher representation theory
Andrea
Appel
University of Edinburgh, UK
Ilknur
Egilmez
University of Southern California, Los Angeles, USA
Matthew
Hogancamp
University of Southern California, Los Angeles, USA
Aaron
Lauda
University of Southern California, Los Angeles, USA
Symmetric functions, nilHecke algebra, categorification, cohomology of Grassmannian
We investigate analogs of symmetric functions arising from an extension of the nilHecke algebra defined by Naisse and Vaz. These extended symmetric functions form a subalgebra of the polynomial ring tensored with an exterior algebra.We define families of bases for this algebra and show that it admits a family of differentials making it a sub-DG-algebra of the extended nilHecke algebra. The ring of extended symmetric functions equipped with this differential is quasi-isomorphic to the cohomology of a Grassmannian. We also introduce new deformed differentials on the extended nilHecke algebra that when restricted makes extended symmetric functions quasi-isomorphic to $GL(N)$-equivariant cohomology of Grassmannians.
Associative rings and algebras
Combinatorics
169
214
10.4171/JCA/2-2-3
http://www.ems-ph.org/doi/10.4171/JCA/2-2-3
5
8
2018
3
Root operators, root groups and retractions
Petra
Schwer
Karlsruhe Institute of Technology, Germany
Path model, root operators, root groups, buildings, retractions
We prove that the Gaussent–Littelmann root operators on galleries can be expressed purely in terms of retractions of a (Bruhat–Tits) building. In addition we establish a connection to the root datum at infinity.
Group theory and generalizations
Geometry
215
230
10.4171/JCA/2-3-1
http://www.ems-ph.org/doi/10.4171/JCA/2-3-1
8
14
2018
Cartwright–Sturmfels ideals associated to graphs and linear spaces
Aldo
Conca
Università di Genova, Italy
Emanuela
De Negri
Università di Genova, Italy
Elisa
Gorla
Université de Neuchâtel, Switzerland
Gröbner basis, generic Initial Ideal, multiview ideals, binomial edge ideals
Inspired by work of Cartwright and Sturmfels, in [15] we introduced two classes of multigraded ideals named after them. These ideals are defined in terms of properties of their multigraded generic initial ideals. The goal of this paper is showing that three families of ideals that have recently attracted the attention of researchers are Cartwright–Sturmfels ideals. More specifically, we prove that binomial edge ideals, multigraded homogenizations of linear spaces, and multiview ideals are Cartwright–Sturmfels ideals, hence recovering and extending recent results of Herzog, Hibi, Hreinsdottir, Kahle, and Rauh [20], Ohtani [28], Ardila and Boocher [3], Aholt, Sturmfels, and Thomas [2], and Binglin Li [6].We also propose a conjecture on the rigidity of local cohomology modules of Cartwright–Sturmfels ideals, that was inspired by a theorem of Brion. We provide some evidence for the conjecture by proving it in the monomial case.
Commutative rings and algebras
Combinatorics
Algebraic geometry
Computer science
231
257
10.4171/JCA/2-3-2
http://www.ems-ph.org/doi/10.4171/JCA/2-3-2
8
14
2018
The monodromy of real Bethe vectors for the Gaudin model
Noah
White
University of California, Los Angeles, USA
Cactus group, Bethe ansatz, Schubert intersections, coboundary categories, crystals
The Bethe algebras for the Gaudin model act on the multiplicity space of tensor products of irreducible $\mathfrak{gl}_r$-modules and have simple spectrum over real points. This fact is proved by Mukhin, Tarasov and Varchenko who also develop a relationship to Schubert intersections over real points. We use an extension to $\overline{M}_{0,n+1}(\mathbb{R})$ of these Schubert intersections, constructed by Speyer, to calculate the monodromy of the spectrum of the Bethe algebras. We show this monodromy is described by the action of the cactus group $J_n$ on tensor products of irreducible $\mathfrak{gl}_r$-crystals.
Nonassociative rings and algebras
Combinatorics
259
300
10.4171/JCA/2-3-3
http://www.ems-ph.org/doi/10.4171/JCA/2-3-3
8
14
2018
A partial order on bipartitions from the generalized Springer correspondence
Jianqiao
Xia
Massachusetts Institute of Technology, Cambridge, USA
Spin group, generalized Springer correspondence, bipartitions
In [1], Lusztig gives an explicit formula for the bijection between the set of bipartitions and the set $\mathcal{N}$ of unipotent classes in a spin group which carry irreducible local systems equivariant for the spin group but not equivariant for the special orthogonal group. The set $\mathcal{N}$ has a natural partial order and therefore induces a partial order on bipartitions. We use the explicit formula given in [1] to prove that this partial order on bipartitions is the same as the dominance order appeared in Dipper–James–Murphy's work [2].
Group theory and generalizations
301
309
10.4171/JCA/2-3-4
http://www.ems-ph.org/doi/10.4171/JCA/2-3-4
8
14
2018
4
Polynomially-bounded Dehn functions of groups
Alexander
Olshanskii
Vanderbilt University, Nashville, USA, and Moscow State University, Russia
Generators and relations in groups, finitely presented group, Dehn function of group, isoperimetric spectrum, Turing machine, S-machine, van Kampen diagram
It is well known that every subqadratic Dehn function is linear. A question by Bridson asked to describe the isoperimetric spectrum of groups, that is the set of all numbers such that $n^{\alpha}$ is equivalent to the Dehn function of a finitely presented group. The goal of this paper is to give a description of the isoperimetric spectrum. Earlier a similar description was given by Sapir, Birget and Rips for the intersection of the isoperimetric spectrum with $[4, \infty]. Lowering the bound from 4 to 2 required significant new ideas and tools.
Group theory and generalizations
Mathematical logic and foundations
311
433
10.4171/JCA/2-4-1
http://www.ems-ph.org/doi/10.4171/JCA/2-4-1
10
25
2018
The isoperimetric spectrum of finitely presented groups
Mark
Sapir
Vanderbilt University, Nashville, USA
Dehn function, finitely presented group, isoperimetric spectrum, $\mathbf {P = NP}$
The isoperimeric spectrum consists of all real positive numbers $\alpha$ such that $n^{\alpha}$ is equivalent to the Dehn function of a finitely presented group. In this note we show how a recent result of Olshanskii completes the description of the isoperimetric spectrum modulo the celebrated Computer Science conjecture (and one of the seven Millennium Problems) $\mathbf {P = NP}$ and even a formally weaker conjecture.
Group theory and generalizations
Mathematical logic and foundations
435
441
10.4171/JCA/2-4-2
http://www.ems-ph.org/doi/10.4171/JCA/2-4-2
10
25
2018