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4d$$ \mathcal{N} $$ 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.

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.

We relate the category of sheaves on alcoves that was constructed in "Sheaves on the alcoves and modular representations I" to the representation theory of reductive algebraic groups. In particular, we show that its indecomposable projective objects encode the simple rational characters of a reductive algebraic group in all characteristics above the Coxeter number., 38 pages. Some corrections in version 3. This is the companion article to "Sheaves on the alcoves and modular representations I". Both articles replace our earlier submissions arXiv:1508.05579 and arXiv:1504.01699

We introduce bijections between generalized type An noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley–Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley–Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on n + 1 strands. We then show that our bijections come from triangular base changes between the diagram basis and the various bases indexed by noncrossing partitions, by explicitly describing the orders giving triangularity. These orders were introduced in a joint paper with Williams and provide exotic lattice structures on noncrossing partitions. Several combinatorial objects are introduced along the way, including an involution on the set of noncrossing partitions.

Let G be a simply connected simple algebraic group over C , B and B − be its two opposite Borel subgroups. For two elements u, v of the Weyl group W, it is known that the coordinate ring C [ G u , v ] of the double Bruhat cell G u , v = B u B ∩ B − v B − is isomorphic to a cluster algebra A ( i ) C [2] , [12] . In the case u = e , v = c 2 (c is a Coxeter element), the algebra C [ G e , c 2 ] has only finitely many cluster variables. In this article, for G = SL r + 1 ( C ) , we obtain explicit forms of all the cluster variables in C [ G e , c 2 ] by considering its additive categorification via preprojective algebras, and describe them in terms of monomial realizations of Demazure crystals.

Let G ˜ = GL ( n , F ¯ q ) where q is a power of a prime. Let F be the standard Frobenius map of G ˜ . Let B ˜ denote the Borel subgroup of G ˜ of upper triangular matrices in G ˜ and let T ˜ denote the maximal split torus of G ˜ contained in B ˜ . Let W denote the Weyl group of the B ˜ , N ˜ pair where N ˜ = N G ˜ ( T ˜ ) . Let w ∈ W and let w ˙ denote an element in the pre-image of the natural surjection map from N G ˜ ( T ˜ ) to W. The double coset B ˜ w ˙ B ˜ contains only regular elements if and only if w is a Coxeter element of minimal length. In the following we consider an element w ∈ W that is not of minimal length and thus B ˜ w B ˜ contains nonregular elements. We explicitly describe certain cosets that are subsets of B ˜ w B ˜ and contain only regular elements. These cosets are chosen for their importance in the construction of a basis of a Hecke algebra associated with G = G ˜ F .

For a classical group G and a Coxeter element c of the Weyl group, it is known that the coordinate ring ℂ [ G e , c 2 ] of the double Bruhat cell G e , c 2 ≔ B ∩ B − c 2 B − has a structure of cluster algebra of finite type, where B and B − are opposite Borel subgroups. In this article, we consider the case G is of type B r , C r or D r and describe all the cluster variables in ℂ [ G e , c 2 ] as monomial realizations of certain Demazure crystals.

For a symmetrizable Kac–Moody Lie algebra $${\mathfrak {g}}$$ , we construct a family of weighted quivers $$Q_m({\mathfrak {g}})$$ ( $$m \ge 2$$ ) whose cluster modular group $$\Gamma _{Q_m({\mathfrak {g}})}$$ contains the Weyl group $$W({\mathfrak {g}})$$ as a subgroup. We compute explicit formulae for the corresponding cluster $${{\mathcal {A}} }$$ - and $${{\mathcal {X}} }$$ -transformations. As a result, we obtain green sequences and the cluster Donaldson–Thomas transformation for $$Q_m({\mathfrak {g}})$$ in a systematic way when $${\mathfrak {g}}$$ is of finite type. Moreover if $${\mathfrak {g}}$$ is of classical finite type with the Coxeter number h, the quiver $$Q_{kh}({\mathfrak {g}})$$ ( $$k \ge 1$$ ) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmuller space of a once-punctured disk with 2k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmuller space of a general marked surface. We finally prove that this action coincides with the one constructed in Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint.

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, and we establish a bijection between these lattices and 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., 17 pages

Though iterated maps and dynamical systems are not new to combinatorics, they have enjoyed a renewed prominence over the past decade through the elevation of the subfield that has become known as dynamical algebraic combinatorics. Some of the problems that have gained popularity can also be cast and analyzed as finite asynchronous cellular automata (CA). However, these two fields are fairly separate, and while there are some individuals who work in both, that is the exception rather than the norm. In this article, we will describe our ongoing work on toggling independent sets on graphs. This will be preceded by an overview of how this project arose from new combinatorial problems involving homomesy, toggling, and resonance. Though the techniques that we explore are directly applicable to ECA rule 1, many of them can be generalized to other cellular automata. Moreover, some of the ideas that we borrow from cellular automata can be adapted to problems in dynamical algebraic combinatorics. It is our hope that this article will inspire new problems in both fields and connections between them., OASIcs, Vol. 90, 27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021), pages 5:1-5:16

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 "vertically and horizontally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily finite type). 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., 31 pages, 10 figures

Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when $p$ is sufficiently large. We also prove variants of these statements involving "relative support varieties.", 54 pages, 2 color figures. v3: minor corrections and additions

Let A = K Q / I be a finite dimensional triangular K -algebra. Let ϕ A be the Coxeter matrix of A . We relate homological conditions for A with properties of the traces of the Coxeter transformation ϕ A . For instance, a finite dimensional accessible algebra A is strongly accessible if and only if Tr ( ϕ A ) = − 1 . We say A is of cyclotomic type if the eigenvalues of ϕ A lie on the unit circle. Clearly, if A is of cyclotomic type then | Tr ( ϕ A ) k | ≤ n , for k ≥ 0 . We prove that A is of cyclotomic type if | Tr ( ϕ A ) k | ≤ n holds for 0 ≤ k ≤ n . We illustrate the results with examples of Nakayama algebras.

We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N=(2, 0) theory of type $J$ on a punctured sphere. We denote the AD theories as $(J^b[k],Y)$, where $J^b[k]$ and $Y$ represent an irregular and a regular singularity respectively. We restrict to the `minimal' case where $J^b[k]$ has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the W-algebra $\mathcal{W}^{k_{2d}}(J,Y)$, where $k_{2d}=-h+ \frac{b}{b+k}$ with $h$ being the dual Coxeter number of $J$. We verify this conjecture by showing that the Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the Hall-Littlewood index computes the Hilbert series of the Higgs branch. We also find that the Schur and Hall-Littlewood index for the AD theory can be written in a simple closed form for $b=h$. We also test the conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5-brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations., Comment: 35 pages, 1 figure, v2: minor corrections, referenced added

We give an explicit description for a weight three generator of the coset vertex operator algebra $C_{L_{\widehat{\sl_{n}}}(l,0)\otimes~L_{\widehat{\sl_{n}}}(1,0)}(L_{\widehat{\sl_{n}}}(l+1,0))$, for $n\geq~2$, $l\geq~1$. Furthermore, we prove that the commutant $C_{L_{\widehat{\sl_{3}}}(l,0)\otimes~L_{\widehat{\sl_{3}}}(1,0)}(L_{\widehat{\sl_{3}}}(l+1,0))$ is isomorphic to the $\W$-algebra $\W_{-3+\frac{l+3}{l+4}}(\sl_3)$, which confirms the conjecture for the $\sl_3$ casethat $C_{L_{\widehat{\frak~g}}(l,0)\otimes~L_{\widehat{\frak~g}}(1,0)}(L_{\widehat{\frak~g}}(l+1,0))$ is isomorphic to$\W_{-h+\frac{l+h}{l+h+1}}(\frak~g)$ for simply-laced Lie algebras ${\frak~g}$ with its Coxeter number $h$ for a positive integer $l$.

In this paper we give a description of the structure of the based ring of the lowest two-sided cell for a weighted Coxeter group with complete graph. It is proved that this ring is generated, in a simple way, by some subrings which are isomorphic to the based rings of the lowest two-sided cells for affine Weyl groups of type A ˜ 1 or A ˜ 2 .

The main algorithmic problems of group theory posed by M. Dehn are the problem of words, the problem of the conjugation of words for finitely presented groups, and the group’s isomorphism problem. Among the works related to the study of the M. Dehn’s problems, the most outstanding ones are the work of P. S. Novikov who proved the undecidability of the problem of words and the conjugacy problem for finitely presented groups as well as the undecidability of the problem of isomorphism of groups. In this regard, the main algorithmic problems and their various generalizations are studied in certain classes of groups. Coxeter groups were introduced by H. S. M. Coxeter: every reflection group is a Coxeter group if its generating elements are reflections with respect to hyperplanes limiting its fundamental polyhedron. H. S. M. Coxeter listed all the reflection groups in three-dimensional Euclidean space and proved that they are all Coxeter groups and every finite Coxeter group is isomorphic to some reflection group in the three-dimensional Euclidean space which elements have a common fixed point. In an algebraic aspect Coxeter groups are studied starting with works by J. Tits who solved the problem of words in certain Coxeter groups. The article describes the known results obtained in solving algorithmic problems in Coxeter groups; the main purpose of the paper is to analyze of the results of solving algorithmic problems in Coxeter groups that were obtained by members of the Tula algebraic school ’Algorithmic problems of theory of the groups and semigroups ’ under the supervision of V. N. Bezverkhnii. It reviews assertions and theorems proved by the authors of the article for the various classes of Coxeter groups: Coxeter groups of large and extra-large types, Coxeter groups with a tree-structure, and Coxeter groups with n-angled structure. The basic approaches and methods of evidence among which the method of diagrams worked out by van Kampen, reopened by R. Lindon and refined by V. N. Bezverkhnii concerning the introduction of R-cancellations, special R-cancellations, special ring cancellations as well as method of graphs, method of types worked out by V. N. Bezverkhnii, method of special set of words designed by V. N. Bezverkhnii on the basis of the generalization of Nielsen method for free construction of groups. Classes of group considered in the article include all Coxeter groups which may be represented as generalized tree structures of Coxeter groups formed from Coxeter groups with tree structure with replacing some vertices of the corresponding tree-graph by Coxeter groups of large or extra-large types as well as Coxeter groups with n-angled structure.

We prove a characterization of irreducible, non-spherical and non-affine Coxeter groups, motivated by applications to metric commensurators of Kac-Moody groups and twinnings at finite distance.

We present new efficient data structures for elements of Coxeter groups of type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually, elements of $H(A_m)$ are represented as simple coefficient list of length $M = (m+1)!$ with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of $H(A_m)$ are represented as nested coefficient lists. While the cost of addition is the same in both data structures, the new data structure leads to a huge improvement in the cost of multiplication in~$H(A_m)$., Comment: 17 pages; errors corrected; example added. To appear in J. Symb. Comp

Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with Frobenius kernel $G_{(1)}$. It is known that when $p\ge 2h-2$, where $h$ is the Coxeter number of $G$, the projective indecomposable $G_{(1)}$-modules (PIMs) lift to $G$, and this has been conjectured to hold in all characteristics. In this paper, we explore the lifting problem via extensions of algebraic groups, following the work of Parshall and Scott who in turn build upon ideas due to Donkin. We prove various results which augment this approach, and as an application demonstrate that the PIMs lift to $G_{(1)}H$, for particular closed subgroups $H \le G$ which contain a maximal torus of $G$., Comment: 13 pages, v2. comments very welcome

Let G be a simple algebraic group of adjoint type over $${\mathbb {C}},$$ whose root system is of type $$F_{4}.$$ Let T be a maximal torus of G and B be a Borel subgroup of G containing T. Let w be an element of the Weyl group W and X(w) be the Schubert variety in the flag variety G/B corresponding to w. Let $$Z(w, {\underline{i}})$$ be the Bott–Samelson–Demazure–Hansen variety (the desingularization of X(w)) corresponding to a reduced expression $${\underline{i}}$$ of w. In this article, we study the cohomology modules of the tangent bundle on $$Z(w_{0}, {\underline{i}}),$$ where $$w_{0}$$ is the longest element of the Weyl group W. We describe all the reduced expressions of $$w_{0}$$ in terms of a Coxeter element such that $$Z(w_{0}, {\underline{i}})$$ is rigid (see Theorem 7.1). Further, if G is of type $$G_{2},$$ there is no reduced expression $${\underline{i}}$$ of $$w_{0}$$ for which $$Z(w_{0}, {\underline{i}})$$ is rigid (see Theorem 8.2).

We prove the conjecture by Gyenge, N\'emethi and Szendr\H{o}i in arXiv:1512.06844, arXiv:1512.06848 giving a formula of the generating function of Euler numbers of Hilbert schemes of points $\operatorname{Hilb}^n(\mathbb C^2/\Gamma)$ on a simple singularity $\mathbb C^2/\Gamma$, where $\Gamma$ is a finite subgroup of $\mathrm{SL}(2)$. We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with $\Gamma$ at $\zeta = \exp(\frac{2\pi i}{2(h^\vee+1)})$ are always $1$, which is a special case of a conjecture by Kuniba [Kun93]. Here $h^\vee$ is the dual Coxeter number. We also prove the claim, which was not known for $E_7$, $E_8$ before., Comment: 17 pages

For a Coxeter element $c$ of a finite Coxeter group, we consider a family of subword complexes parameterized by reduced expressions of the longest element. This family generalizes $c-$cluster complexes. We describe vertices of these complexes in terms of roots of the corresponding root system. We prove that dual polytopes of all such complexes are combinatorial 2-truncated cubes., Comment: Short announcement note published in 2014. Details will appear in forthcoming work

We consider a class of trigonometric solutions of WDVV 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 $\vee$-system and we show that their subsystems are also trigonometric $\vee$-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., Comment: 32 pages