Dynkin Type Research Articles

Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras

For an element w of a simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory of the module category over the preprojective algebra of Dynkin type. This paper studies categorical properties of using the root system. We show that simple objects in bijectively correspond to Bruhat inversion roots of w, and obtain a combinatorial criterion for to satisfy the Jordan-Hölder property (JHP). For type A case, we give a diagrammatic construction of simple objects, and show that (JHP) can be characterized via a forest-like permutation, introduced by Bousquet-Mélou and Butler in the study of Schubert varieties.

Tensor Product Multiplicities via Upper Cluster Algebras

For each valued quiver $Q$ of Dynkin type, we construct a valued ice quiver $\Delta_Q^2$. Let $G$ be a simple connected Lie group with Dynkin diagram the underlying valued graph of $Q$. The upper cluster algebra of $\Delta_Q^2$ is graded by the triple dominant weights $(\mu,\nu,\lambda)$ of $G$. We prove that when $G$ is simply-laced, the dimension of each graded component counts the tensor multiplicity $c_{\mu,\nu}^\lambda$. We conjecture that this is also true if $G$ is not simply-laced, and sketch a possible approach. Using this construction, we improve Berenstein-Zelevinsky's model, or in some sense generalize Knutson-Tao's hive model in type $A$.

A Coxeter spectral classification of positive edge-bipartite graphs II. Dynkin type [formula omitted

We continue the Coxeter spectral study of finite positive edge-bipartite signed (multi)graphs Δ (bigraphs, for short), with n≥2 vertices started in Simson (2013) [44] and developed in Simson (2018) [49]. We do it by means of the non-symmetric Gram matrix GˇΔ∈Mn(Z) defining Δ, its Gram quadratic form qΔ:Zn→Z, v↦v⋅GˇΔ⋅vtr (that is positive definite, by definition), the complex spectrum speccΔ⊂S1:={z∈C,|z|=1} of the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Mn(Z), called the Coxeter spectrum of Δ, and the Coxeter polynomial coxΔ(t):=det⁡(t⋅E−CoxΔ)∈Z[t]. One of the aims of the Coxeter spectral analysis is to classify the connected bigraphs Δ with n≥2 vertices up to the ℓ-weak Gram Z-congruence Δ∼ℓZΔ′ and up to the strong Gram Z-congruence Δ≈ZΔ′, where Δ∼ℓZΔ′ (resp. Δ≈ZΔ′) means that det⁡GˇΔ=det⁡GˇΔ′ and GΔ′=Btr⋅GΔ⋅B (resp. GˇΔ′=Btr⋅GˇΔ⋅B), for some B∈Mn(Z) with det⁡B=±1, where GΔ:=12[GˇΔ+GˇΔtr]∈Mn(12Z).Here we study connected signed simple graphs Δ, with n≥2 vertices, that are positive, i.e., the symmetric Gram matrix GΔ∈Mn(12Z) of Δ is positive definite. It is known that every such a signed graph is ℓ-weak Gram Z-congruent with a unique simply laced Dynkin graph DynΔ∈{An,Dn,n≥4,E6,E7,E8}, called the Dynkin type of Δ. A classification up to the strong Gram Z-congruence Δ≈ZΔ′ is still an open problem and only partial results are known. In this paper, we obtain such a classification for the positive signed simple graphs Δ of Dynkin type Dn by means of the family of the signed graphs Dn(1)=Dn,Dn(2),…,Dn(rn) constructed in Section 2, for any n≥4, where rn=⌞n/2⌟. More precisely, we prove that any connected signed simple graph of Dynkin type Dn, with n≥4 vertices, is strongly Z-congruent with a signed graph Dn(s), for some s≤rn, and its Coxeter polynomial coxΔ(t) is of the form (ts+1)(tn−s+1). We do it by a matrix morsification type reduction to the classification of the conjugacy classes in the integral orthogonal group O(n,Z) of the integer orthogonal matrices C, with det⁡(E−C)=4.

THE MINIMAL MODULAR FORM ON QUATERNIONIC

AbstractSuppose that $G$ is a simple reductive group over $\mathbf{Q}$, with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$-valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$, which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$. We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$, which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$.

Quadratic algorithm to compute the Dynkin type of a positive definite quasi-Cartan matrix

Cartan matrices and quasi-Cartan matrices play an important role in such areas as Lie theory, representation theory, and algebraic graph theory. It is known that each (connected) positive definite quasi-Cartan matrix A ∈ M n ( Z ) A\in \mathbb {M}_n(\mathbb {Z}) is Z \mathbb {Z} -equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of A A . We present a symbolic, graph-theoretic algorithm to compute the Dynkin type of A A , of the pessimistic arithmetic (word) complexity O ( n 2 ) \mathcal {O}(n^2) , significantly improving the existing algorithms. As an application we note that our algorithm can be used as a positive definiteness test for an arbitrary quasi-Cartan matrix, more efficient than standard tests. Moreover, we apply the algorithm to study a class of (symmetric and non-symmetric) quasi-Cartan matrices related to Nakayama algebras.

Gröbner–Shirshov Basis of Derived Hall Algebra of Type An

We know that in Ringel-Hall algebra of Dynkin type, the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis, and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra. We aim to generalize this result to the derived Hall algebra DH(An) of type An. First, we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category Db(An) using the Auslander-Reiten quiver of Db(An), and then we prove that all possible compositions between these skew commutator relations are trivial. As an application, we give a PBW type basis of DH(An).

Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$

We build a bijection between the set $\sttilt\Lambda$ of isomorphism classes of basic support $\tau$-tilting modules over the Auslander algebra $\Lambda$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\sttilt\Lambda$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tilt\Lambda$ of isomorphism classes of basic tilting $\Lambda$-modules and the symmetric group $\mathfrak{S}_n$ due to Br\"{u}stle, Hille, Ringel and R\"{o}hrle. Regarding the preprojective algebra $\Gamma$ of Dynkin type $A_n$ as a factor algebra of $\Lambda$, we show that the tensor functor $-\otimes_{\Lambda}\Gamma$ induces a bijection between $\sttilt\Lambda\to\sttilt\Gamma$. This recover Mizuno's bijection $\mathfrak{S}_{n+1}\to\sttilt\Gamma$ for type $A_n$.

A geometric q‐character formula for snake modules

Let $\mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of a simple complex Lie algebra ${\mathfrak{g}}$. Let $\mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. We prove the geometric $q$-character formula conjectured by Hernandez and Leclerc in types $\mathbb{A}$ and $\mathbb{B}$ for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the $F$-polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category $\mathscr{C}_1$ are factorial for Dynkin types $\mathbb{A,D,E}$.

The Fourier expansion of modular forms on quaternionic exceptional groups

Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G(\mathbf{R})$. The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$, along the unipotent radical $N$ of the Heisenberg parabolic $P = MN$ of $G$.

Stability and indecomposability of the representations of quivers of An-type

In his paper, Markus Reineke proposed a conjecture that there exists a stable weight system Θ for every indecomposable representation of Dynkin type quiver. In this paper, we showed this conjecture is true for quivers of An-type by combinatorial construction of a special weight system. We also reinterpret this weight system in terms of semi-invariant theory.

Frieze vectors and unitary friezes

Let Q be a quiver without loops and 2-cycles, let A(Q) be the corresponding cluster algebra and let x be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to x. These frieze vectors are defined as solutions of certain Diophantine equations given by the cluster variables in the cluster algebra. We show that every cluster gives rise to a frieze vector and that the frieze vector determines the cluster. We also study friezes of type Q as homomorphisms from the cluster algebra to an arbitrary integral domain. In particular, we show that every positive integral frieze of affine Dynkin type A is unitary, which means it is obtained by specializing each cluster variable in one cluster to the constant 1. This completes the answer to the question of unitarity for all positive integral friezes of Dynkin and affine Dynkin types.

On polynomial time inflation algorithm for loop-free non-negative edge-bipartite graphs

We study a class of signed graphs called finite connected loop-free edge-bipartite graphs Δ (bigraphs, for short), started in Simson (2013) and continued in Simson and Zając (2017) and Makuracki and Simson (2019). In this paper we present an algorithmic approach to the study of non-negative bigraphs Δ with n+r≥1 vertices Δ0={a1,…,an+r} of corank r≥0, that is, bigraphs with the symmetric Gram matrix GΔ≔12(GˇΔ+GˇΔtr)∈Mn+r(12Z) positive semi-definite of rank n≥1, where GˇΔ∈Mn+r(Z) is the non-symmetric Gram matrix of Δ.One of the main results of this paper asserts that every such a bigraph Δ with n+r≥1 vertices is algorithmically reduced by using at most O(n2) inflation operators tab−, a,b∈Δ0, to one of the canonical r-vertex extensions D̂n(r)∈{Ân(r),D̂n(r),Ê6(r),Ê7(r),Ê8(r)} of the simply laced Dynkin diagram Dn∈{An,Dn,E6,E7,E8}, where Dn is the Dynkin type of Δ, that is, a unique simply laced Dynkin graph Dn=DynΔ∈{An,Dn,E6,E7,E8} associated with Δ. We also construct inflation algorithm that: (a) computes in quadratic time a Dynkin type DynΔ of Δ, and (b) constructs a matrix B∈Mn+r(Z), with detB=±1, defining the weak Gram Z-congruence Δ∼ZBD̂n(r), that is, satisfying the equation GD̂n(r)=Btr⋅GΔ⋅B.

Three results for $\tau $-rigid modules

$\tau $-rigid modules are essential in the $\tau $-tilting theory introduced by Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms of $\tau $-rigid modules. We show that every indecomposable module over iterated tilted algebras of Dynkin type is $\tau $-rigid. Finally, we give a $\tau $-tilting theorem on homological dimension which is an analog to that of classical tilting modules.

A computational technique in Coxeter spectral study of symmetrizable integer Cartan matrices

With any symmetrizable integer Cartan matrix C∈SCarn⊆Mn(Z), a Z-invertible Coxeter matrix CoxC∈Mn(Z) is associated. We study such positive definite matrices up to a strong Gram Z-congruence C≈ZC′ (defined in the paper by means of a Z-invertible matrix B∈Mn(Z)) by means of the Dynkin type DynC, the complex spectrum speccC⊂C of the Coxeter polynomial coxC(t):=det⁡(tE−CoxC)∈Z[t] and the Coxeter type CtypC=(speccC,swC) of C. We show that the strong Gram Z-congruence C≈ZC′ implies the equality CtypC=CtypC′. The inverse implication is an open problem studied in the paper. However, we prove the implication for positive definite symmetrizable integer Cartan matrices C,C′ satisfying any of the following two conditions: (i) if n≤9, the matrices C,C′ are symmetric (i.e., swC=swC′=1) and DynC is one of the simply laced Dynkin graphs An, n≥1, Dn, n≥4, E6, E7, E8, and (ii) n≥2, the matrices C,C′ are not symmetric and DynC is one of the Dynkin signed graphs Bn, Cn, F4, G2. We do it by a reduction to an analogous problem for positive Cox-regular edge-bipartite graphs.Moreover, given n≤9, we present a list of 60 pairwise non-congruent matrices in SCarn such that any positive definite irreducible symmetrizable integer Cartan matrix in Mn(Z), with n≤9, is strongly Gram Z-congruent with a matrix of the list, see Theorem 4.3.

Factoriality and class groups of cluster algebras

Locally acyclic cluster algebras are Krull domains. Hence their factorization theory is determined by their (divisor) class group and the set of classes containing height-1 prime ideals. Motivated by this, we investigate class groups of cluster algebras. We show that any cluster algebra that is a Krull domain has a finitely generated free abelian class group, and that every class contains infinitely many height-1 prime ideals. For a cluster algebra associated to an acyclic seed, we give an explicit description of the class group in terms of the initial exchange matrix. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type. In the acyclic case, we prove the sufficiency of necessary conditions for factoriality given by Geiss–Leclerc–Schröer.

Fusion hierarchies, T-systems and Y-systems for the dilute loop models

The fusion hierarchy, T-system and Y-system of functional equations are the key to exact solvability for 2d lattice models. We derive these equations for the generic dilute loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of . For generic values of the crossing parameter , the T- and Y-systems do not truncate. For the case rational so that is a root of unity, we find explicit closure relations and derive closed finite T- and Y-systems. The thermodynamics Bethe ansatz (TBA) integral equations and the associated TBA diagrams of the Y-systems are not of simple Dynkin type. They involve nodes if p is even and nodes if p is odd and are related to the TBA diagrams of models at roots of unity by a folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are . Prototypical examples of the loop models, at roots of unity, include critical dense polymers with central charge c = −2, and loop fugacity and critical site percolation on the triangular lattice with c = 0, and . Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their counterparts. More specifically, it will confirm the extent to which bond and site percolation lie in the same universality class as logarithmic conformal field theories.

Semistable subcategories for tiling algebras

Semistable subcategories were introduced in the context of Mumford’s GIT and interpreted by King in terms of representation theory of finite dimensional algebras. Ingalls and Thomas later showed that for finite dimensional algebras of Dynkin and affine type, the poset of semistable subcategories is isomorphic to the corresponding poset of noncrossing partitions. We show that semistable subcategories defined by tiling algebras, introduced by Coelho Simoes and Parsons, are in bijection with noncrossing tree partitions, introduced by the second author and McConville. Moreover, this bijection defines an isomorphism of the posets on these objects. Our work recovers that of Ingalls and Thomas in Dynkin type A.

Representation theoretic realization of non-symmetric Macdonald polynomials at infinity

Abstract We study the non-symmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules for simply-laced Dynkin types except for type E 8 {\mathrm{E}_{8}} .

Autonomous limit of the 4-dimensional Painlevé-type equations and degeneration of curves of genus two

In recent studies, 4-dimensional analogs of the Painlevé equations were listed and there are 40 types. The aim of the present paper is to geometrically characterize these 40 Painlevé-type equations. For this purpose, we study the autonomous limit of these equations and degeneration of their spectral curves. The spectral curves are 2-parameter families of genus two curves and their generic degeneration are one of the types classified by Namikawa and Ueno. Liu’s algorithm enables us to find the degeneration types of the spectral curves for our 40 types of integrable systems. This result is analogous to the following fact; the families of the spectral curves of the autonomous 2-dimensional Painlevé equations P I , P II , P IV , P III D 8 , P III D 7 , P III D 6 , P V and P VI define elliptic surfaces with the singular fiber at H=∞ of the Dynkin types E 8 (1) , E 7 (1) , E 6 (1) , D 8 (1) , D 7 (1) , D 6 (1) , D 5 (1) and D 4 (1) , respectively.

On the Radical of the Module Category of an Endomorphism Algebra

Given a finite dimensional algebra A over an algebraically closed field we study the relationship between the powers of the radical of a morphism in the module category of the algebra A and the induced morphism in the module category of the endomorphism algebra of a tilting A-module. We compare the nilpotency indices of the radical of the mentioned module categories. We find an upper bound for the nilpotency index of the radical of the module category of iterated tilted algebras of Dynkin type.