Double Bruhat Cells Research Articles

Representations of shifted quantum affine algebras and cluster algebras I: The simply laced case

AbstractWe introduce a family of cluster algebras of infinite rank associated with root systems of type , , . We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called ‐system. We conjecture that all cluster monomials are classes of simple objects of . In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply connected algebraic group. This explains the similarity between ‐system relations and certain generalized minor identities discovered by Fomin and Zelevinsky.

MV polytopes and reduced double Bruhat cells

When G is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of G . By fixing w from the Weyl group, we can define MV polytopes whose highest vertex is labelled by w . We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by w^{-1} . To do this, we define a collection of generalized minor functions \Delta_{\gamma}^{\mathrm{new}} which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex w . We also describe the combinatorial structure of MV polytopes of highest vertex w . We explicitly describe the map from the Weyl group to the subset of elements bounded by w in the Bruhat order which sends u \mapsto v if the vertex labelled by u coincides with the vertex labelled by v for every MV polytope of highest vertex w . As a consequence of this map, we prove that these polytopes have vertices labelled by Weyl group elements less than w in the Bruhat order.

Cluster algebra structures on Poisson nilpotent algebras

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.

Purity and Separation for Oriented Matroids

Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in [ n ] [n] has the same cardinality. In this paper, we extend these notions and define M \mathcal {M} -separated collections for any oriented matroid M \mathcal {M} . We show that maximal by size M \mathcal {M} -separated collections are in bijection with fine zonotopal tilings (if M \mathcal {M} is a realizable oriented matroid), or with one-element liftings of M \mathcal {M} in general position (for an arbitrary oriented matroid). We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid M \mathcal {M} is pure if M \mathcal {M} -separated collections form a pure simplicial complex, i.e., any maximal by inclusion M \mathcal {M} -separated collection is also maximal by size. We pay closer attention to several special classes of oriented matroids: oriented matroids of rank 3 3 , graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank 3 3 is pure if and only if it is a positroid (up to reorienting and relabeling its ground set). A graphical oriented matroid is pure if and only if its underlying graph is an outerplanar graph, that is, a subgraph of a triangulation of an n n -gon. We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank 3 3 , graphical, uniform).

Relative cluster categories and Higgs categories

Cluster categories were introduced in 2006 by Buan-Marsh-Reineke-Reiten-Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot (2009) [3] and Plamondon (2011) [50] to arbitrary cluster algebras associated with quivers. A higher dimensional generalization is due to Guo (2011) [17]. Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, etc. The work of Geiss-Leclerc-Schröer often yields Frobenius exact categories which allow us to categorify such cluster algebras.In this paper, we generalize the construction of (higher) cluster categories by Claire Amiot and by Lingyan Guo to the relative context. We prove the existence of an n-cluster tilting object in a Frobenius extriangulated category, namely the Higgs category (generalizing the Frobenius categories of Geiss-Leclerc-Schröer), which is stably n-Calabi–Yau and Hom-finite, arising from a left (n+1)-Calabi–Yau morphism. Our results apply in particular to relative Ginzburg dg algebras coming from ice quivers with potential and higher Auslander algebras associated to n-representation-finite algebras.

Q-opers,QQ-systems, and Bethe Ansatz II: Generalized minors

AbstractIn this paper, we describe a certain kind ofq-connections on a projective line, namelyZ-twisted(G,q){(G,q)}-opers with regular singularities using the language of generalized minors. In part one we explored the correspondence between theseq-connections and𝑄𝑄\mathit{QQ}-systems/Bethe Ansatz equations. Here we associate to aZ-twisted(G,q){(G,q)}-oper a class of meromorphic sections of aG-bundle, satisfying certain difference equations, which we refer to as(G,q){(G,q)}-Wronskians. Among other things, we show that the𝑄𝑄\mathit{QQ}-systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.

Bases for upper cluster algebras and tropical points

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of their bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland’s representation-theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds.

Configuration Poisson Groupoids of Flags

Abstract Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in ${\mathcal {B}}^n$. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

Generalized Minors and Tensor Invariants

Abstract In the paper [1], the authors define functions on double Bruhat cells, which they call generalized minors. By relating certain double Bruhat cells to the spaces $\operatorname {Conf}_3 {\mathcal {A}}_G$ and $\operatorname {Conf}_4 {\mathcal {A}}_G$, we give formulas for these generalized minors as tensor invariants. This allows us to verify certain weight identities conjectured in [7]. We then show that the grading of these tensor invariants is equivalent information to the quiver for the cluster structure on $\operatorname {Conf}_3 {\mathcal {A}}_G$. This leads to a simple combinatorial construction of cluster structures on the moduli space of framed local systems ${\mathcal {X}}_{G^{\prime},S}$ and ${\mathcal {A}}_{G,S}$.

Redundancy in string cone inequalities and multiplicities in potential functions on cluster varieties

We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence.

Langlands duality and Poisson–Lie duality via cluster theory and tropicalization

Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to G: its Langlands dual group \(G^\vee \), and its Poisson–Lie dual group \(G^*\), respectively. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein–Kazhdan potential on the double Bruhat cell \(G^{\vee ; w_0, e} \subset G^\vee \) is isomorphic to the integral Bohr–Sommerfeld cone defined by the Poisson structure on the partial tropicalization of \(K^* \subset G^*\) (the Poisson–Lie dual of the compact form \(K \subset G\)). By Berenstein and Kazhdan (in: Contemporary mathematics, vol. 433. American Mathematical Society, Providence, pp 13–88, 2007), the first cone parametrizes the canonical bases of irreducible G-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of \(K^*\) are equal to symplectic volumes of the corresponding coadjoint orbits in \({{\,\mathrm{Lie}\,}}(K)^*\). To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by Fock and Goncharov (Ann Sci Ec Norm Supér (4) 42(6):865–930, 2009). These are pairs of cluster varieties whose seed matrices are transpose to each other. There is a naturally defined isomorphism between their tropicalizations. The isomorphism between the cones described above is a particular instance of such an isomorphism associated to the double Bruhat cells \(G^{w_0, e} \subset G\) and \(G^{\vee ; w_0, e} \subset G^\vee \).

Solution of tetrahedron equation and cluster algebras

We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Given a standard complex semisimple Poisson Lie group $(G, \pi_{st})$, generalised double Bruhat cells $G^{u, v}$ and generalised Bruhat cells $O^u$ equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang Hua Lu and the author. We prove in this paper that $G^{u,u}$ is naturally a Poisson groupoid over $O^u$, extending a result from the aforementioned authors about double Bruhat cells in $(G, \pi_{st})$. Our result on $G^{u,u}$ is obtained as an application of a construction interesting in its own right, of a local Poisson groupoid over a mixed product Poisson structure associated to the action of a pair of Lie bialgebras. This construction involves using a local Lagrangian bisection in a double symplectic groupoid closely related to the global R-matrix studied by Weinstein and Xu, to twist a direct product of Poisson groupoids.

Polyhedral parametrizations of canonical bases & cluster duality

We establish the relation of Berenstein–Kazhdan's decoration function and Gross–Hacking–Keel–Kontsevich's potential on the open double Bruhat cell in the base affine space G/N of a simple, simply connected, simply laced algebraic group G. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on G/N arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations. In the appendix we construct maximal green sequences for the open double Bruhat cell in G/N which is a crucial assumption for Gross–Hacking–Keel–Kontsevich's construction.

The Berenstein–Zelevinsky quantum cluster algebra conjecture

We prove the Berenstein–Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite-dimensional connected, simply connected simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [5]. We furthermore prove that the corresponding upper quantum cluster algebras coincide with the constructed quantum cluster algebras and exhibit a large number of explicit quantum seeds. Along the way a detailed study of the properties of quantum double Bruhat cells from the viewpoint of noncommutative UFDs is carried out and a quantum analog of the Fomin–Zelevinsky twist map is constructed and investigated for all double Bruhat cells. The results are valid over base fields of arbitrary characteristic and the deformation parameter is only assumed to be a non-root-of-unity.

The twist for positroids

There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu ̈cker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.

Donaldson-Thomas transformation of double bruhat cells in semisimple Lie groups

Double Bruhat cells $G^{u,v}$ were studied by Fomin and Zelevinsky. They provide important examples of cluster algebras and cluster Poisson varieties. Cluster varieties produce examples of 3d Calabi-Yau categories with stability conditions, and their Donaldson-Thomas invariants, defined by Kontsevich and Soibelman, are encoded by a formal automorphism on the cluster variety known as the Donaldson-Thomas transformation. Goncharov and Shen conjectured in that for any semisimple Lie group $G$, the Donaldson-Thomas transformation of the cluster Poisson variety $H\backslash G^{u,v}/H$ is a slight modification of Fomin and Zelevinsky's twist map. In this paper we prove this conjecture, using crucially Fock and Goncharov's cluster ensembles and the amalgamation construction. Our result, combined with the work of Gross, Hacking, Keel, and Kontsevich, proves the duality conjecture of Fock and Goncharov in the case of $H\backslash G^{u,v}/H$.

Geometric crystals and cluster ensembles in Kac–Moody setting

For a Kac–Moody group G, double Bruhat cells Gu,e (u is a Weyl group element) have positive geometric crystal structures. In Williams (2013), it is shown that there exist birational maps between ‘cluster tori’ XΣ (resp. AΣ) and GAdu,e (resp. Gu,e), and they are extended to regular maps from cluster X (resp. A) -varieties to GAdu,e (resp. Gu,e). The aim of this article is to compute explicit formulae for the induced geometric crystal structures on these cluster tori. As a corollary, the sets of ZT-valued points of these cluster varieties have crystal structures.

Cluster algebras of finite type via Coxeter elements and Demazure crystals of type A

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[Gu,v] of the double Bruhat cell Gu,v=BuB∩B−vB− is isomorphic to a cluster algebra A(i)C[2,12]. In the case u=e, v=c2 (c is a Coxeter element), the algebra C[Ge,c2] has only finitely many cluster variables. In this article, for G=SLr+1(C), we obtain explicit forms of all the cluster variables in C[Ge,c2] by considering its additive categorification via preprojective algebras, and describe them in terms of monomial realizations of Demazure crystals.

Affine cluster monomials are generalized minors

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with$\mathbf{g}$-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type$A_{1}^{(1)}$, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.