Coxeter Number Research Articles

Domain walls in 4d mathcal{N} = 1 SYM

4d mathcal{N} = 1 super Yang-Mills (SYM) with simply connected gauge group G has h gapped vacua arising from the spontaneously broken discrete R-symmetry, where h is the dual Coxeter number of G. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We put forward an explicit answer to this question for all the domain walls for G = SU(N), Sp(N), Spin(N) and G2, and for the minimal domain wall connecting neighboring vacua for arbitrary G. We propose that the domain wall theories support specific nontrivial topological quantum field theories (TQFTs), which include the Chern-Simons theory proposed long ago by Acharya-Vafa for SU(N). We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions twisted by global symmetries of SYM computed in the ultraviolet with those computed in our proposed infrared TQFTs. A crucial element in this matching is constructing the Hilbert space of spin TQFTs, that is, theories that depend on the spin structure of spacetime and admit fermionic states — a subject we delve into in some detail.

Birational Rowmotion and Coxeter-Motion on Minuscule Posets

Birational rowmotion is a discrete dynamical system on the set of all positive real-valued functions on a finite poset, which is a birational lift of combinatorial rowmotion on order ideals. It is known that combinatorial rowmotion for a minuscule poset has order equal to the Coxeter number, and exhibits the file homomesy phenomenon for refined order ideal cardinality statistics. In this paper we generalize these results to the birational setting. Moreover, as a generalization of birational promotion on a product of two chains, we introduce birational Coxeter-motion on minuscule posets, and prove that it enjoys periodicity and file homomesy.

Picture groups and maximal green sequences

<p style='text-indent:20px;'>We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).</p><p style='text-indent:20px;'>Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.</p>

Braid groups of type ADE, Garside monoids, and the categorified root lattice

We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$, a baric structure on $\mathcal{K}$. We use these baric structures to define metrics on the braid group, and we identify these metrics with the word-length metrics in the Birman-Ko-Lee/Bessis dual generators of the associated dual positive monoid $\mathbb{B}_{W.c}^\vee$. As consequences, we give new proofs that the standard and dual positive monoids inject into the group, give linear-algebraic solutions to the membership problem in the standard and dual positive monoids, and provide new proofs of the faithfulness of the action of $\mathbb{B}_W$ on $\mathcal{K}$. Finally, we use the compatibility of the baric and $t$-structures on $\mathcal{K}$ to prove a conjecture of Digne and Gobet regarding the canonical word-length of the dual simple generators of ADE braid groups.

Trigonometric -systems and solutions of WDVV equations * *In memory of Jonathan Nimmo

We consider a class of trigonometric solutions of Witten–Dijkgraaf–Verlinde–Verlinde equations determined by collections of vectors with multiplicities. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find new solutions given by restrictions of root systems, as well as examples which are not of this form. Further, we consider a closely related notion of a trigonometric ∨-system, and we show that its subsystems are also trigonometric ∨-systems. Finally, while reviewing the root system case we determine a version of (generalised) Coxeter number for the exterior square of the reflection representation of a Weyl group.

Nilpotent Elements and Reductive Subgroups Over a Local Field

Let ${\mathcal {K}}$ be a local field – i.e. the field of fractions of a complete DVR ${\mathcal {A}}$ whose residue field k has characteristic p > 0 – and let G be a connected, absolutely simple algebraic ${\mathcal {K}}$ -group G which splits over an unramified extension of ${\mathcal {K}}$ . We study the rational nilpotent orbits of G– i.e. the orbits of $G({\mathcal {K}})$ in the nilpotent elements of $\text {Lie}(G)({\mathcal {K}})$ – under the assumption p > 2h − 2 where h is the Coxeter number of G. A reductive group M over ${\mathcal {K}}$ is unramified if there is a reductive model ${{\mathscr{M}}}$ over ${\mathcal {A}}$ for which $M = {{\mathscr{M}}}_{{\mathcal {K}}}$ . Our main result shows for any nilpotent element X1 ∈Lie(G) that there is an unramified, reductive ${\mathcal {K}}$ -subgroup M which contains a maximal torus of G and for which X1 ∈Lie(M) is geometrically distinguished. The proof uses a variation on a result of DeBacker relating the nilpotent orbits of G with the nilpotent orbits of the reductive quotient of the special fiber for the various parahoric group schemes associated with G.

The Largest Coefficient of the Highest Root and the Second Smallest Exponent

There are many different ways that the exponents of Weyl groups of irreducible root systems have been defined and put into practice. One of the most classical and algebraic definitions of the exponents is related to the eigenvalues of Coxeter elements. While the coefficients of the height root when expressed as a linear combination of simple roots are combinatorial objects in nature, there are several results asserting relations between these exponents and coefficients. This study was conducted to give a uniform and fairly elementary proof of the fact that the second smallest exponent of the Weyl group is one or two plus the largest coefficient of the highest root of the root system depending upon a simple condition on the root lengths. As a consequence, we obtain a necessary and sufficient condition for a root system to be of type $G_2$ in terms of these numbers.

Lattice bijections for string modules, snake graphs and the weak Bruhat order

In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding abstract snake graph. In particular, we make explicit the direct correspondence between a submodule of a string module and the perfect matching of the corresponding snake graph. For every string module we define a Coxeter element in a symmetric group. We then establish a bijection between the submodule lattice of the string module and the lattice given by the interval in the weak Bruhat order determined by the Coxeter element. Using the correspondence between string modules and snake graphs, we give a new concise formulation of snake graph calculus.

A Note on the Hurwitz Action on Reflection Factorizations of Coxeter Elements in Complex Reflection Groups

We show that the Hurwitz action is "as transitive as possible" on reflection factorizations of Coxeter elements in the well-generated complex reflection groups $G(d,1,n)$ (the group of $d$-colored permutations) and $G(d,d,n)$.

Dual Garside structures and Coxeter sortable elements

In Artin-Tits groups attached to Coxeter groups of spherical type, we give a combinatorial formula to express the simple elements of the dual braid monoids in the classical Artin generators. Every simple dual braid is obtained by lifting an $S$-reduced expression of its image in the Coxeter group, in a way which involves Reading's $c$-sortable elements. It has as an immediate consequence that simple dual braids are Mikado braids (the known proofs of this result either require topological realizations of the Artin groups or categorification techniques), and hence that their images in the Iwahori-Hecke algebras have positivity properties. In the classical types, this requires to give an explicit description of the inverse of Reading's bijection from $c$-sortable elements to noncrossing partitions of a Coxeter element $c$, which might be of independent interest. The bijections are described in terms of the noncrossing partition models in these types. While the proof of the formula is case-by-case, it is entirely combinatorial and we develop an approach which reduces a uniform proof to uniformly proving a lemma about inversion sets of $c$-sortable elements.

Rigid connections on P1 via the Bruhat–Tits building

We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid $G$-connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy-Prasad filtration associated to a point in the Bruhat-Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus $S$, then we have an $S$-toral connection. In this language, the irregular singularity of the Frenkel-Gross connection gives rise to the homogenous toral connection of minimal slope associated to the Coxeter torus $\mathcal{C}$. In the present paper, we consider connections on $\mathbb{G}_m$ which have an irregular homogeneous $\mathcal{C}$-toral singularity at zero of slope $i/h$, where $h$ is the Coxeter number and $i$ is a positive integer coprime to $h$, and a regular singularity at infinity with unipotent monodromy. Our main result is the characterisation of all such connections which are rigid.

SHEAVES ON THE ALCOVES AND MODULAR REPRESENTATIONS I

We consider the set of affine alcoves associated with a root system R as a topological space and consider a certain category S of sheaves of Z-modules on this space. Here Z is the structure algebra of the root system over a field k. To any wall reflection we associate a wall crossing functor on S. In the companion article Sheaves on the alcoves and modular representations II we prove that S encodes the simple rational characters of the connected, simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number.

On character formulas for simple and tilting modules

We show that the characters of tilting modules can be used, in a concrete and explicit way, to obtain the simple characters of a connected reductive algebraic group G over an algebraically closed field k of characteristic p, for all p. Thus, once an algorithm for computing the characters of the indecomposable tilting G-modules has been found, an algorithm for the simple characters has been also. An immediate implication is that work by Achar, Makisumi, Riche, and Williamson can be used to obtain the characters for simple G-modules when p>h, the Coxeter number of G.

SHEAVES ON THE ALCOVES AND MODULAR REPRESENTATIONS II

We consider the set of affine alcoves associated with a root system R as a topological space and define a certain category S of sheaves of $$ {\mathcal{Z}}_k $$ -modules on this space. Here $$ {\mathcal{Z}}_k $$ is the structure algebra of the root system over a field k. To any wall reection s we associate a wall crossing functor on S. In the companion article [FL] we prove that S encodes the simple rational characters of the connected, semisimple and simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number.

The action of a Coxeter element on an affine root system

The characterization of orbits of roots under the action of a Coxeter element is a fundamental tool in the study of finite root systems and their reflection groups. This paper develops the analogous tool in the affine setting, adding detail and uniformity to a result of Dlab and Ringel.

Factorization problems in complex reflection groups

AbstractWe enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of W-noncrossing partitions.

VOAs and Rank-Two Instanton SCFTs

We analyze the mathcal {N}=2 superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number. We comment briefly on expectations for the still higher-rank cases.

Exceptional Cycles in the Bounded Derived Categories of Quivers

An exceptional n-cycle in a Horn-finite triangulated category with Serre functor has been recently introduced by Broomhead, Pauksztello and Ploog. When n = 1, it is a spherical object. We explicitly determine all the exceptional cycles in the bounded derived category Db (kQ) of a finite quiver Q without oriented cycles. In particular, if Q is an Euclidean quiver, then the length type of exceptional cycles in Db (kQ) is exactly the tubular type of Q; if Q is a Dynkin quiver of type Em (m = 6, 7, 8), or Q is a wild quiver, then there are no exceptional cycles in Db (kQ); and if Q is a Dynkin quiver of type An or Dn, then the length of an exceptional cycle in Db (kQ) is either h or $$\frac{h}{2}$$, where h is the Coxeter number of Q.

Coxeter exponents and orthogonal complements of highest roots

In this note, we study the subroot system of an irreducible (reduced, crystallographic) root system obtained by taking the orthogonal complement of the highest root. We describe root theoretic data (the Coxeter number and the largest coefficient of the highest root when expressed in terms of the simple roots) of this subroot system in terms of that of As an application, we obtain a new description of the Coxeter exponents of a complex simple Lie algebra.

Truncation and the Induction Theorem

A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg compares the bounded derived category Db(block(U)) of finite dimensional modules for the principal block of a Lusztig quantum enveloping algebra U at an ℓth root of unity with a special full subcategory Dtriv(B) of the bounded derived category of integrable type 1 modules for a Borel part B⊂U. Specifically, according to this “Induction Theorem” [1, Theorem 3.5.5] the right derived functor of induction IndBU yields an equivalence of categories RIndBU:Dtriv(B)→∼Db(block(U)). Some restrictions on ℓ are required–e.g., ℓ>h, the Coxeter number. It is suggested briefly [1, Remark 3.5.6] that an analog of this equivalence carries over to characteristic p>0 representations of algebraic groups. Indeed, the authors of the present paper have verified, in a separate preprint [6], that there is such an equivalence RIndBG:Dtriv(B)→∼Db(block(G)) relating an analog of Dtriv(B), defined using a Borel subgroup B of a simply connected semisimple algebraic group G, to the bounded derived category of the principal block of finite dimensional rational G-modules. The proof is not without difficulty and supplies new, previously missing details even in the quantum case. The present paper continues the study of the modular case, taking the derived category equivalence as a starting point. The main result here is that, assuming p>2h−2, the equivalence behaves well with respect to certain weight poset “truncations,” making use of a variation by Woodcock [12] on van der Kallen's “excellent order” [10]. This means, in particular, that the equivalence can be reformulated in terms of derived categories of finite dimensional quasi-hereditary algebras. We expect that a similar result holds in the quantum case.In [9, Thm. 2.1] J. Rickard proves a theorem stating existence, in an algebraic groups context, of some derived equivalences with specified corresponding character isometries. In an appendix to this paper, we discuss a similar more recent result [6, Lem. 3.2]. It gives additional information on the behavior of the derived equivalences, giving their action on some natural right-derived induced objects, lifting the given character isometries.