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.

HALF POTENTIAL ON GEOMETRIC CRYSTALS AND CONNECTEDNESS OF CELLULAR CRYSTALS

For any simply connected, simple complex algebraic group, we define upper/lower half-decorated geometric crystals and show that their tropicalization will be upper/lower normal Kashiwara's crystals. In particular, we show that the tropicalization of the half-decorated geometric crystal on the big Bruhat cell \( \left(=B{\overline{w}}_0:= {B}^{-}\cap U{\overline{w}}_0U\right) \) is isomorphic to the Langlands dual crystal B(∞) of the nilpotent-half subalgebra of quantum group. As an application, we shall show that any cellular crystal associated with a reduced word is connected in the sense of a crystal graph.

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 $G$.

A Bruhat Atlas for the Mehta–van der Kallen Stratification of T∗GLn/B

Mehta and van der Kallen put a Frobenius splitting on the type A cotangent bundle T∗GLn/B, thereby defining a stratification by compatibly split subvarieties, and they determined a few of the elements of this stratification. We embed T∗GLn/B as a stratum in a larger stratified (and Frobenius split) space GLn/B × Matn whose stratification we determine, thereby giving a full description of the one of Mehta–van der Kallen. The main technique is to endow GLn/B × Matn with a Bruhat atlas, covering it with open sets that are stratified-isomorphic to Bruhat cells (in GL2n/B2n). Among the consequences are that each stratum closure is normal and Cohen–Macaulay.

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.

On the Homotopy Type of Intersections of Two Real Bruhat Cells

Abstract Real Bruhat cells give an important and well-studied stratification of such spaces as $\operatorname {GL}_{n+1}$, $\operatorname {Flag}_{n+1} = \operatorname {SL}_{n+1}/B$, $\operatorname {SO}_{n+1}$, and $\operatorname {Spin}_{n+1}$. We study the intersections of a top dimensional cell with another cell (for another basis). Such an intersection is naturally identified with a subset of the lower nilpotent group $\operatorname {Lo}_{n+1}^{1}$. We are particularly interested in the homotopy type of such intersections. In this paper, we define a stratification of such intersections. As a consequence, we obtain a finite CW complex that is homotopically equivalent to the intersection. We compute the homotopy type for several examples. It turns out that for $n \le 4$ all connected components of such subsets of $\operatorname {Lo}_{n+1}^1$ are contractible: we prove this by explicitly constructing the corresponding CW complexes. Conversely, for $n \ge 5$ and the top permutation, there is always a connected component with even Euler characteristic and therefore not contractible. This follows from formulas for the number of cells per dimension of the corresponding CW complex. For instance, for the top permutation $S_6$, there exists a connected component with Euler characteristic equal to $2$. We also give an example of a permutation in $S_6$ for which there exists a connected component that is homotopically equivalent to the circle ${{\mathbb {S}}}^1$.

Factorizations of $k$-nonnegative matrices

A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set of generators and relations for the semigroup of $k$-nonnegative $n\times n$ invertible matrices in two special cases: when $k = n-1$ and when $k = n-2$, restricted to unitriangular matrices. For these two cases, we prove that the set of $k$-nonnegative matrices can be partitioned into cells based on their factorizations into generators, generalizing the notion of Bruhat cells from totally nonnegative matrices. Like Bruhat cells, these cells are homeomorphic to open balls and have a topological structure that neatly relates closure of cells to subwords of factorizations. In the case of $(n-2)$-nonnegative unitriangular matrices, we show the cells form a Bruhat-like CW-complex.

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.

Locally convex curves and the Bruhat stratification of the spin group

We study the lifting of the Schubert stratification of the homogeneous space of complete real flags of $R^{n+1}$ to its universal covering group $Spin_{n+1}$. We call the lifted strata the Bruhat cells of $Spin_{n+1}$, in keeping with the homonymous classical decomposition of reductive algebraic groups. We present explicit parameterizations for these Bruhat cells in terms of minimal-length expressions $\sigma=a_{i_1}... a_{i_k}$ for permutations $\sigma\in S_{n+1}$ in terms of the $n$ generators $a_i=(i,i+1)$. These parameterizations are compatible with the Bruhat orders in the Coxeter-Weyl group $S_{n+1}$. This stratification is an important tool in the study of locally convex curves; we present a few such applications.

The universal valuation of Coxeter matroids

Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag varieties. In the process, we compute the universal valuation of generalized Coxeter permutohedra, a larger family of polyhedra that model Coxeter analogues of combinatorial objects such as matroids, clusters, and posets.

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.

The Full Symmetric Toda Flow and Intersections of Bruhat Cells

In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements $w$, $w'$ in the Weyl group $W(\mathfrak g)$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w'}$ iff $w\prec w'$ in the Bruhat order on $W(\mathfrak g)$. Here $\mathfrak g$ is a normal real form of a semisimple complex Lie algebra $\mathfrak g_\mathbb C$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations.