Blob Algebra Research Articles

Annular webs and Levi subalgebras

For any Levi subalgebra of the form \mathfrak{l}=\mathfrak{gl}_{l_{1}}\oplus\cdots\oplus\mathfrak{gl}_{l_{d}}\subseteq\mathfrak{gl}_{n} we construct a quotient of the category of annular quantum \mathfrak{gl}_{n} webs that is equivalent to the category of finite-dimensional representations of quantum \mathfrak{l} generated by exterior powers of the vector representation. This can be interpreted as an annular version of skew Howe duality, gives a description of the representation category of \mathfrak{l} by additive idempotent completion, and a web version of the generalized blob algebra.

Path isomorphisms between quiver Hecke and diagrammatic Bott–Samelson endomorphism algebras

We construct an explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson's diagrammatic endomorphism algebras of Bott–Samelson bimodules. As a corollary, we deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan–Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. We hence give an elementary and more explicit proof of the main theorem of Riche–Williamson's recent monograph and extend their categorical equivalence to cyclotomic quiver Hecke algebras, thus solving Libedinsky–Plaza's categorical blob conjecture.

{U}_{mathfrak{q}}{mathfrak{sl}}_2 -invariant non-compact boundary conditions for the XXZ spin chain

We introduce new {U}_{mathfrak{q}}{mathfrak{sl}}_2 -invariant boundary conditions for the open XXZ spin chain. For generic values of mathfrak{q} we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for mathfrak{q}={e}^{frac{ipi}{p}} a 2p-th root of unity — to its p-dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ.To motivate our construction, we first specialise to mathfrak{q} = i, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the (η, ξ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations.Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if mathfrak{q} α ∉± mathfrak{q} ℤ, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to {U}_{mathfrak{q}}{mathfrak{sl}}_2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of mathfrak{q} and α. Finally, we conjecture the general form of the Schur-Weyl duality in this case.This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

Handlebody diagram algebras

In this paper we study handlebody versions of some classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer, BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory.

Tensor product categorifications, Verma modules and the blob 2-category

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal \mathfrak{sl}_2 Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorical action of \mathfrak{sl}_2 . From the above we derive a categorification of the blob algebra.

Schur–Weyl duality, Verma modules, and row quotients of Ariki–Koike algebras

We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of $\mathfrak{gl}_m$. We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particular case of our algebras.

Type [formula omitted] Temperley-Lieb algebra quotients and Catalan combinatorics

We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type C˜, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.

The nil-blob algebra: An incarnation of type [formula omitted] Soergel calculus and of the truncated blob algebra

We introduce a type B analogue of the nil Temperley-Lieb algebra in terms of generators and relations, that we call the (extended) nil-blob algebra. We show that this algebra is isomorphic to the endomorphism algebra of a Bott-Samelson bimodule in type A˜1. We also prove that it is isomorphic to an idempotent truncation of the classical blob algebra.

Indecomposable tilting modules for the blob algebra

The blob algebra is a finite-dimensional quotient of the Hecke algebra of type B which is almost always quasi-hereditary. We construct the indecomposable tilting modules for the blob algebra over a field of characteristic 0 in the doubly critical case. Every indecomposable tilting module of maximal highest weight is either a projective module or an extension of a simple module by a projective module. Moreover, every indecomposable tilting module is a submodule of an indecomposable tilting module of maximal highest weight. We conclude that the graded Weyl filtration multiplicities of the indecomposable tilting modules in this case are given by inverse Kazhdan–Lusztig polynomials of type A˜1.

EXOTIC SPRINGER FIBERS FOR ORBITS CORRESPONDING TO ONE-ROW BIPARTITIONS

We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato's original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the one-boundary Temperley-Lieb algebra (also known as the blob algebra). This provides a first step in generalizing the geometric versions of Khovanov's arc algebra to the exotic setting.

Blob algebra approach to modular representation theory

Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type $\tilde{A}_1$. In this paper we take that observation far beyond its original scope. We conjecture that for $\tilde{A}_n$ there is an equivalence of categories between the characteristic $p$ diagrammatic Hecke category and a "blob category" that we introduce (using certain quotients of KLR algebras called \emph{generalized blob algebras}). Using alcove geometry we prove the "graded degree" part of this equivalence for all $n$ and all prime numbers $p$. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic $p$ give the $p$-Kazhdan Lusztig polynomials in type $\tilde{A}_n$. We prove this for $\tilde{A}_1$, the only case where the $p$-Kazhdan Lusztig polynomials are known.

Graded cellular basis and Jucys-Murphy elements for generalized blob algebras

We give a concrete construction of a graded cellular basis for the generalized blob algebra Bn introduced by Martin and Woodcock. The construction uses the isomorphism between KLR-algebras and cyclotomic Hecke algebras, proved by Brundan-Kleshchev and Rouquier. It gives rise to a family of Jucys-Murphy elements for Bn.

Blob algebra and two-color Soergel calculus

In 2003, Martin and Woodcock noticed a connection between the representation theory of the blob algebra and the Kazhdan–Lusztig polynomials associated with the infinite dihedral group. However, no conceptual explanation for this coincidence has yet been provided. In this study, a possible explanation of this phenomenon is suggested by enunciating a conjecture that relates the endomorphism algebra of Bott–Samelson bimodules to certain subalgebras of the blob algebra obtained by idempotent truncation. Evidence supporting this conjecture is provided.

On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories

The relationship between bulk and boundary properties is one of the founding features of (rational) conformal field theory (CFT). Our goal in this paper is to explore the possibility of having an equivalent relationship in the context of lattice models. We focus on models based on the Temperley–Lieb algebra, and use the concept of ‘braid translation’, which is a natural way, in physical terms, to ‘close’ an open spin chain by adding an interaction between the first and last spins using braiding to ‘bring’ them next to each other. The interaction thus obtained is in general non-local, but has the key feature that it is expressed solely in terms of the algebra for the open spin chain—the ‘ordinary’ Temperley–Lieb algebra and its blob algebra generalization. This is in contrast with the usual periodic spin chains which involve only local interactions, and are described by the periodic Temperley–Lieb algebra. We show that for the restricted solid-on-solid models, which are known to be described by minimal unitary CFTs (with central charge ) in the continuum limit, the braid translation in fact does provide the ordinary periodic model starting from the open model with fixed (identical) boundary conditions on the two sides of the strip. This statement has a precise mathematical formulation, which is a pull-back map between irreducible modules of, respectively, the blob algebra and the affine Temperley–Lieb algebra. We then turn to the same kind of analysis for two models whose continuum limits are logarithmic CFTs (LCFTs)—the alternating and spin chains. We find that the result for minimal models does not hold any longer: braid translation of the relevant (in that case, indecomposable but not irreducible) modules of the Temperley–Lieb algebra does not give rise to the modules known to be present in the periodic chains. In the case, the content in terms of the irreducibles is the same, as well as the spectrum, but the detailed structure (like logarithmic coupling) is profoundly different. This carries over to the continuum limit. The situation is similar for the case. The problem of relating bulk and boundary lattice models for LCFTs thus remains open.

Graded cellular bases for Temperley–Lieb algebras of type A and B

We show that the Temperley---Lieb algebra of type A and the blob algebra (also known as the Temperley---Lieb algebra of type B) at roots of unity are $\mathbb{Z}$ -graded algebras. We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.

Graded decomposition numbers for the blob algebra

We give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity.

A physical approach to the classification of indecomposable Virasoro representations from the blob algebra

In the context of Conformal Field Theory (CFT), many results can be obtained from the representation theory of the Virasoro algebra. While the interest in Logarithmic CFTs has been growing recently, the Virasoro representations corresponding to these quantum field theories remain dauntingly complicated, thus hindering our understanding of various critical phenomena. We extend in this paper the construction of Read and Saleur (2007) [1,2], and uncover a deep relationship between the Virasoro algebra and a finite-dimensional algebra characterizing the properties of two-dimensional statistical models, the so-called blob algebra (a proper extension of the Temperley–Lieb algebra). This allows us to explore vast classes of Virasoro representations (projective, tilting, generalized staggered modules, etc.), and to conjecture a classification of all possible indecomposable Virasoro modules (with, in particular, L0 Jordan cells of arbitrary rank) that may appear in a consistent physical Logarithmic CFT where Virasoro is the maximal local chiral algebra. As by-products, we solve and analyze algebraically quantum-group symmetric XXZ spin chains and sl(2|1) supersymmetric spin chains with extra spins at the boundary, together with the “mirror” spin chain introduced by Martin and Woodcock (2003) [3].

Cell structures on the blob algebra

We consider ther=0r = 0case of the conjectures by Bonnafé, Geck, Iancu and Lam on cellular structures on the Hecke algebra of typeBB. We show that this case induces the natural cell structure on the blob algebrabnb_nby restriction to one-line bipartitions.

A PRESENTATION FOR THE SYMPLECTIC BLOB ALGEBRA

The symplectic blob algebra bn(n ∈ ℕ) is a finite-dimensional algebra defined by a multiplication rule on a basis of certain diagrams. The rank r(n) of bn is not known in general, but r(n)/n grows unboundedly with n. For each bn we define an algebra by presentation, such that the number of generators and relations grows linearly with n. We prove that these algebras are isomorphic.

Edge states and conformal boundary conditions in super spin chains and super sigma models

The sigma models on projective superspaces CP N + M − 1 | N with topological angle θ = π mod 2 π flow to non-unitary, logarithmic conformal field theories in the low-energy limit. In this paper, we determine the exact spectrum of these theories for all open boundary conditions preserving the full global symmetry of the model, generalizing recent work on the particular case M = 0 [C. Candu et al., JHEP 1002 (2010) 015]. In the sigma model setting, these boundary conditions are associated with complex line bundles, and are labelled by an integer, related with the exact value of θ. Our approach relies on a spin chain regularization, where the boundary conditions now correspond to the introduction of additional edge states. The exact values of the exponents then follow from a lengthy algebraic analysis, a reformulation of the spin chain in terms of crossing and non-crossing loops (represented as a certain subalgebra of the Brauer algebra), and earlier results on the so-called one- and two-boundary Temperley–Lieb algebras (also known as blob algebras). A remarkable result is that the exponents, in general, turn out to be irrational. The case M = 1 has direct applications to the spin quantum Hall effect, which will be discussed in a sequel.