We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen-Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\mathrm{Ext}^1$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.
This short note is an extended abstract for my talk at the Nottingham Online Algebraic Geometry Seminar on October 1, 2020. It is based on the joint works with Alexander Kuznetsov arXiv:1802.08097 and arXiv:2001.04148.
In this paper we study extensions between Cohen-Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine ${\rm Ext}^i(L_I, L_J)$ for arbitrary rank 1 modules $L_I$ and $L_J$. An explicit combinatorial algorithm is given for computation of ${\rm Ext}^i(L_I, L_J)$ when $i$ is odd, and for $i$ even, we show that ${\rm Ext}^i(L_I, L_J)$ is cyclic over the centre, and we give an expli…
The category of Cohen-Macaulay modules of an algebra $B_{k,n}$ is used [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander--Reiten sequences and study the Auslander--Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen-Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the assoc…
We associate a dimer algebra A to a Postnikov diagram D (in a disk) corresponding to a cluster of minors in the cluster structure of the Grassmannian Gr(k,n). We show that A is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay module T over the algebra B used to categorify the cluster structure of Gr(k,n) by Jensen-King-Su. It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disk. The construction and proof uses an interpretation of the diagram D, with its associated plabic g…
In this paper we compute the generating rank of $k$-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of $k$-Grassmannians arising from Hermitian forms of Witt index $n$ defined over vector spaces of dimension $N > 2n$. We also study generating sets for the $2$-Grassmannians arising from quadratic forms of Witt index $n$ defined over $V(N,{\mathbb F}_q)$ for $q=4,8,9$ and $2n \leq N \leq 2n+2$. We prove that for $N >6$ they can be generated over the prime subfield, thus determining their generating rank.
For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional $\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra. Properties like graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.
In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $A_n$ and $D_n$, i.e., flag varieties $Fl(1,n;n+1)$ and isotropic orthogonal Grassmannians $OG(2,2n)$; in particular we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $OG(2…
Rank $1$ modules are the building blocks of the category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra of affine type $A$. Jensen, King and Su showed in \cite{JKS16} that the category ${\rm CM}(B_{k,n})$ provides an additive categorification of the cluster algebra structure on the coordinate ring $\mathbb C[{\rm Gr}(k, n)]$ of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$. Rank $1$ modules are indecomposable, they are known to be in bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit construction has…
The category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$, \cite{JKS16}. Among the indecomposable modules in this category are the rank $1$ modules which are in bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit construction has been given by Jensen, King and Su. These are the building blocks of the category as any module in ${\rm CM}(B_{k,n}) $ can be filtered by the…
We describe diagrammatically a positively graded Koszul algebra \mathbb{D}_k such that the category of finite dimensional \mathbb{D}_k-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type D_k constructible with respect to the Schubert stratification. The connection is given by an explicit isomorphism to the endomorphism algebra of a projective generator described in by Braden. The algebra is obtained by a "folding" procedure from the generalized Khovanov arc algebras. We relate this algebra to the category of finite dimensional representations of the …
We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cell…
We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G((t))/G[[t]]. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I^0 equivariant categories, where I^0 is the radical of the Iwahori subgroup of G((t)). Our result may be viewed as an affine analogue of the equivalence…
In this paper we study indecomposable rank 2 modules in the Grassmannian cluster category ${\rm CM}(B_{5,10})$. This is the smallest wild case containing modules whose profile layers are $5$-interlacing. We construct all rank 2 indecomposable modules with filtration $\{i,i+2,i+4,i+6,i+8\}\mid \{i+1,i+3,i+5,i+7,i+9\}$, classify them up to isomorphism, and parameterize all infinite families of non-isomorphic rank 2 modules.
In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic $p$ bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application…
In \cite{JKS} we gave an (additive) categorification of Grassmannian cluster algebras, using the category $\CM(A)$ of Cohen-Macaulay modules for a certain Gorenstein order $A$. In this paper, using a cluster tilting object in the same category $\CM(A)$, we construct a compatible pair $(B, L)$, which is the data needed to define a quantum cluster algebra. We show that when $(B, L)$ is defined from a cluster tilting object with rank 1 summands, this quantum cluster algebra is (generically) isomorphic to the corresponding quantum Grassmannian.
We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups.
Kontsevich and Soibelman defined the notion of Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. A family of examples of such categories can be constructed from an arbitrary cluster variety. The corresponding Donaldson-Thomas invariants are encoded by a special formal automorphism of the cluster variety, known as Donaldson-Thomas transformation. Fix two integers $m$ and $n$ with $1<m<m+1<n$. It is known that the configuration space $\mathrm{Conf}_n(\mathbb{P}^{m-1})$, closely related to Grassmannian $\mathrm{Gr}_m(n)$, is a cluster Poisson variety. In this pap…
We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the loop Grassmannian for the Langlands dual group. The equivalence (1)-(2) is an `enhancement' of the known expression for quantum group cohomology in terms of nilpotent variety, due to Ginzburg-Kumar. The equivalence (2)-(3) is a step towards resolving an old mystery surrounding the existense of two completely different realizations of the affine Hecke algebra…
We revisit the geometric description of cluster categories in type E in terms of colored diagonals in a polygon and generalize it to the case of m-cluster categories. As an application, we relate colored diagonals in a polygon to semi-standard Young tableaux, in type E_6,E_7,E_8. This provides a new compatibility description of semi--standard Young tableaux in Grassmannian cluster algebras in type E_6, E_8 and in a sub-cluster algebra of type E_7.
We study convolution algebras associated with Heckman-Opdam polynomials. For root systems of type BC we derive three continuous classes of positive convolution algebras (hypergroups) by interpolating the double coset convolution structures of compact Grassmannians U/K with fixed rank over the real, complex or quaternionic numbers. These convolution algebras are linked to explicit positive product formulas for Heckman-Opdam polynomials of type BC, which occur for certain discrete multiplicities as the spherical functions of U/K. These results complement those of a recent paper by the second aut…
We categorify various finite-type cluster algebras with coefficients using completed orbit categories associated to Frobenius categories. Namely, the Frobenius categories we consider are the categories of finitely generated Gorenstein projective modules over the singular Nakajima category associated to a Dynkin diagram and their standard Frobenius quotients. In particular, we are able to categorify all finite-type skew-symmetric cluster algebras with universal coefficients and finite-type Grassmannian cluster algebras. Along the way, we classify the standard Frobenius models of a certain famil…
The geometric Satake correspondence provides an equivalence of categories between the Satake category of spherical perverse sheaves on the affine Grassmannian and the category of representations of the dual group. In this note, we define a combinatorial version of the Satake category using irreducible components of fibres of the convolution morphism. We then prove an equivalence of coboundary categories between this combinatorial Satake category and the category of crystals of the dual group.
We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we describe explicitly geometric realizations of the Wakimoto modules as Hecke eigen-D-modules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group.
Nakaoka and Palu introduced the notion of extriangulated categories by extracting the similarities between exact categories and triangulated categories. In this paper, we study cotorsion pairs in a Frobenius extriangulated category $\C$. Especially, for a $2$-Calabi-Yau extriangulated category $\C$ with a cluster structure, we describe the cluster substructure in the cotorsion pairs. For rooted cluster algebras arising from $\C$ with cluster tilting objects, we give a one-to-one correspondence between cotorsion pairs in $\C$ and certain pairs of their rooted cluster subalgebras which we call c…
We give new interpretations of the Deligne categories $\underline{Rep}(GL_t)$ and $\underline{Rep}(S_t)$ (and their abelian envelopes) over $\mathbb{C}$ in terms of modular representations of general linear and symmetric groups of large rank in large characteristic. In particular we make sense of the sentence "$\underline{Rep}(S_n)$ is the limit of $Rep(S_{p+n})$ over $\bar{\mathbb{F}}_p$ as $p$ goes to infinity". We then give examples of how to pass results between these different settings.
Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other. In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in p…
We give two proofs of a level-rank duality for braided fusion categories obtained from quantum groups of type $C$ at roots of unity. The first proof uses conformal embeddings, while the second uses a classification of braided fusion categories associated with quantum groups of type $C$ at roots of unity. In addition we give a similar result for non-unitary braided fusion categories quantum groups of types $B$ and $C$ at odd roots of unity.
We prove that integral blocks of parabolic category O associated to the subalgebra gl(m) x gl(n) of gl(m+n) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of 2-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type A.
We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, extending an earlier construction of the authors for k=2.
For an arbitrary commutative ring k and t in k, we construct a 2-functor S_t which sends a tensor category to a new tensor category. By applying it to the representation category of a bialgebra we obtain a family of categories which interpolates the representation categories of the wreath products of the bialgebra. This generalizes the construction of Deligne's category Rep(S_t,k) for representation categories of symmetric groups.
We classify pointed fusion categories C(G, $ω$) up to Morita equivalence for 1 < |G| < 32. Among them, the cases |G| = 2 3 , 2 4 and 3 3 are emphasized. Although the equivalence classes of such categories are not distinguished by their Frobenius-Schur indicators, their categorical Morita equivalence classes are distinguished by the set of the indicators and ribbon twists of their Drinfeld centers. In particular, the modular data are a complete invariant for the modular categories Z(C(G, $ω$)) for |G[< 32. We use the computer algebra package GAP and present codes for treating complex-valued gro…
