In this largely expository article we present an elementary construction of Lusztig's canonical basis in type ADE. The method, which is essentially Lusztig's original approach, is to use the braid group to reduce to rank two calculations. Some of the wonderful properties of the canonical basis are already visible; that it descends to a basis for every highest weight integrable representation, and that it is a crystal basis.
We derive a formula for the entries of the (unitriangular) transition matrices between the standard monomial and dual canonical bases of the irreducible polynomial representations of U_q(gl_n) in terms of Kazhdan-Lusztig polynomials.
We generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based $\textbf{U}^{\imath}$-module and a based $\textbf{U}$-module is a based $\textbf{U}^{\imath}$-module. This is then used to formulate a Kazhdan-Lusztig theory for an arbitrary parabolic BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebras, extending a main result in [BW13] (arXiv:1310.0103).
For arbitrary reduced words we give formulas for the crystal structures on string and Lusztig data of type A as well as the defining inequalities of the corresponding polytopes revealing certain dualities between them.
We present a new characterization of Lusztig's canonical quotient group. We also define a duality map: to a pair consisting of a nilpotent orbit and a conjugacy class in its fundamental group, the map assigns a nilpotent orbit in the Langlands dual Lie algebra.
Given any symmetric Cartan datum, Lusztig has provided a pair of key lemmas to construct the perverse sheaves over the corresponding quiver and the functions of irreducible components over the corresponding preprojective algebra respectively. In the present article, we prove that these two inductive algorithms of Lusztig coincide. Consequently we can define two colored graphs and prove that they are isomorhic. This result finishes the statement that Lusztig's functions of irreducible components are basis of the enveloping algebra and deduces the crystal structure (in the sense of Kashiwara-Sai…
In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters of the irreducible and tilting modules in the BGG category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundan's conjecture and its variants associated to all Borel subalgebras in full generality.
We define a partial order on the set of pairs {(O,C)}, where O is a nilpotent orbit and C is a conjugacy class in Lusztig's canonical quotient of A(O). We then show that there is a unique order-reversing duality map on this set that has certain properties analagous to those of the original Lusztig-Spaltenstein duality map. This generalizes work of E. Sommers. In this revised version, a new approach leads to a strengthened version of Theorem 2, and applications of the results are discussed in greater detail.
In this note, we show that certain dual canonical basis elements of $\mathbb{C}[SL_m]$ are positive when evaluated on $k$-positive matrices, matrices whose minors of size $k \times k$ and smaller are positive. Skandera showed that all dual canonical basis elements of $\mathbb{C}[SL_m]$ can be written in terms of Kazhdan-Lusztig immanants, which were introduced by Rhoades and Skandera. We focus on the basis elements which are expressed in terms of Kazhdan-Lusztig immanants indexed by 1324- and 2143-avoiding permutations. This extends previous work of the authors on Kazhdan-Lusztig immanants and…
The goal of this work is to provide an elementary construction of the canonical basis $\mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to establish its invariance under modified Lusztig's symmetries. To that effect, we obtain a direct characterization of the upper global basis $\mathbf B^{up}$ in terms of a suitable bilinear form and show that $\mathbf B(w)$ is contained in $\mathbf B^{up}$ and its large part is preserved by modified Lusztig's symmetries.
In the present article, a basis of the coordinate algebra of the multi-parameter quantized matrix is constructed by using an elementary method due to Lusztig. The construction depends heavily on an anti-automorphism, the bar action. The exponential nature of the bar action is derived which provides an inductive way to compute the basis elements. By embedding the basis into the dual basis of Lusztig's canonical basis of $U_q(n^-)$, the positivity properties of the basis as well as the positivity properties of the canonical basis of the modified quantum enveloping algebra of type $A$, which has …
The negative part $U^-$ of a quantised enveloping algebra associated to a simple Lie algebra possesses a canonical basis $\mathcal{B}$ with favourable properties. Lusztig has associated a cone to a reduced expression $\mathbf{i}$ for the longest element $w_0$ in the Weyl group of $\mathfrak{g}$, with good properties with respect to monomial elements of $\mathcal{B}$. The first author has associated a subalgebra $A_{\mathbf{i}}$ of $U^-$, compatible with the dual basis $\mathcal{B}^*$, to each reduced expression $\mathbf{i}$. We show that, after a certain twisting, the string parametrisation of…
A lot of recent activity has been directed towards various constructions of "natural" bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig's construction of a canonical basis, and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke algebra. The key ingredient of our approach is a new version of Lusztig's Lemma that we apply to all acyclic quantum cluster algebras. As a result, we construct the "canonical" basis in every such algebra that we call the canonical triangular basis.
The Iwahori-Hecke algebra $\mathcal{H}$ of a Coxeter system $(W,S)$ has a "standard basis" indexed by the elements of $W$ and a "bar involution" given by a certain antilinear map. Together, these form an example of what Webster calls a pre-canonical structure, relative to which the well-known Kazhdan-Lusztig basis of $\mathcal{H}$ is a canonical basis. Lusztig and Vogan have defined a representation of a modified Iwahori-Hecke algebra on the free $\mathbb{Z}[v,v^{-1}]$-module generated by the set of twisted involutions in $W$, and shown that this module has a unique pre-canonical structure sat…
These are the notes for a series of lectures given on the theory of canonical and crystal bases for Hall algebras (for a summer school in Grenoble in 2008). It may be viewed as a follow-up to arXiv:math/0611617. It covers the construction, due to Lusztig, of the canonical bases for the Hall algebra of a quiver Q in terms of a certain category of perverse sheaves over the moduli space of representations of Q. It also contains an exposition of Kashiwara and Saito's geometric construction of the crystal graph in terms of irreducible components of Lusztig's lagrangian in the cotangent bundle to th…
We obtain a family of explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here "polyhedral" means that the multiplicity in question is expressed as the number of lattice points in some convex polytope. Our answers use a new combinatorial concept of $\ii$-trails which resemble Littelmann's paths but seem to be more tractable. We also study combinatorial structure of Lusztig's canonical bases or, equivalently of Kashiwara's global bases. Although Lusztig's and Kashiwara's approach…
We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula" to compute the actions of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirković conjecture for the crystal structure on MV polytopes.…
We study the canonical basis for the negative part of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras associated to two different matrices satisfying certain conditions may coincide. We show that the canonical bases coincide provided that the algebras coincide. We also answer partially a question by Lusztig.
The modified quantized enveloping algebra $\dot{\mathbf{U}}$ has a remarkable canonical basis, which was introduced by Lusztig. In this paper, we give an explicit description of all elements of the canonical basis of $\dot{\mathbf{U}}$ for type $A_{2}$.
A canonical basis in the sense of Lusztig is a basis of a free module over a ring of Laurent polynomials that is invariant under a certain semilinear involution and is obtained from a fixed "standard basis" through a triangular base change matrix with polynomial entries whose constant terms equal the identity matrix. Among the better known examples of canonical bases are the Kazhdan-Lusztig basis of Iwahori-Hecke algebras, Lusztig's canonical basis of quantum groups and the Howlett-Yin basis of induced $W$-graph modules. This paper has two major theoretical goals: First to show that having bas…
Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In this note we calculate the canonical distinguished involutions in the affine Weyl groups of rank <8. We also prove some partial results relating canonical distinguished involutions and Dynkin's diagrams o…
Motivated by a question on the graded rank of the stalks of the canonical sheaf on a Bruhat graph, we lift some equalities concerning (parabolic) Kazhdan-Lusztig polynomials to this moment graph setting. Our proofs hold also in positive characteristic, under some technical assumptions.
Expanding the classical work of Kazhdan-Lusztig, we construct a bar involution and canonical bases on the $q$-Brauer algebra introduced by Wenzl. We define explicit actions of the $q$-Brauer algebra on the tensor spaces, and formulate $\imath$Schur dualities between the $q$-Brauer algebra and the $\imath$quantum groups of type AI and AII respectively.
It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Geiß, Leclerc and Schröer has recently given a parametrization of irreducible components of nilpotent varieties in studying cluster algebras. In this paper we show that their parametrization coincides with Lusztig's parametrization of the canonical basis.
In this paper we show that there is a link between the combinatorics of the canonical basis of a quantized enveloping algebra and the monomial bases of the second author arising from representations of quivers. We prove that some reparametrization functions of the canonical basis, arising from the link between Lusztig's approach to the canonical basis and the string parametrization of the canonical basis, are given on a large region of linearity by linear functions arising from these monomial basis for a quantized enveloping algebra.
In this paper, we study the Kazhdan--Lusztig cells of a Coxeter group $W$ in a ``relative'' setting, with respect to a parabolic subgroup $W_I \subseteq W$. This relies on a factorization of the Kazhdan--Lusztig basis $\{C_w\}$ of the corresponding (multi-parameter) Iwahori--Hecke algebra with respect to $W_I$. We obtain two applications to the ``asymptotic case'' in type $B_n$, as introduced by Bonnafé--Iancu: we show that $\{C_w\}$ is a ``cellular basis'' in the sense of Graham--Lehrer, and we construct an analogue of Lusztig's canonical isomorphism from the Iwahori--Hecke algebra to the gro…
We investigate the affine canonical basis and the monomial basis constructed in [LXZ] in Lusztig's geometric setting. We show that the transition matrix between the two bases is upper triangular with 1's in the diagonal and coefficients in the upper diagonal entries in N[v, v^{-1}]. As a consequence, we show that part of the monomial basis elements give rise to resolutions of support varieties of the affine canonical basis elements as simple perverse sheaves.
We relate the canonical basis of the Fock space representation of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$, as defined by Leclerc and Thibon, to the canonical basis of its restriction to $U_q(\mathfrak{sl}_{n})$, regarded as a based module in the sense of Lusztig. More generally we consider the restriction to any parabolic subalgebra. We deduce results on decomposition numbers and branching coefficients of Schur algebras over fields of positive characteristic, generalising those of Kleshchev and of Tan and Teo.
We introduce the language of multiplier Hopf algebra in the context of positive representations of split real quantum groups, and discuss its applications with a continuous version of Lusztig-Kashiwara's canonical basis, which may provide a key to prove the closure of the positive representations under tensor products, and harmonic analysis of quantized algebra of functions in the sense of locally compact quantum groups.
The lowest two-sided cell of the extended affine Weyl group $W_e$ is the set $\{w \in W_e: w = x \cdot w_0 \cdot z, \text{for some} x,z \in W_e\}$, denoted $W_{(ν)}$. We prove that for any $w \in W_{(ν)}$, the canonical basis element $\C_w$ can be expressed as $\frac{1}{[n]!} χ_λ({\y}) \C_{v_1 w_0} \C_{w_0 v_2}$, where $χ_λ({\y})$ is the character of the irreducible representation of highest weight $λ$ in the Bernstein generators, and $v_1$ and $v_2^{-1}$ are what we call primitive elements. Primitive elements are naturally in bijection with elements of the finite Weyl group $W_f \subseteq W_e…
For the BGG category of $\mathfrak{q}(n)$-modules of half-integer weights, a Kazhdan-Lusztig conjecture à la Brundan is formulated in terms of categorical canonical basis of the $n$th tensor power of the natural representation of the quantum group of type $C$. For the BGG category of $\mathfrak{q}(n)$-modules of congruent non-integral weights, a Kazhdan-Lusztig conjecture is formulated in terms of canonical basis of a mixed tensor of the natural representation and its dual of the quantum group of type $A$. We also establish a character formula for the finite-dimensional irreducible $\mathfrak{…
The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A_1^{(1)}. The examined subalgebra is associated with a terminal module M over the path algebra of the Kronecker quiver via an Weyl group element w of length four. Geiss-Leclerc-Schroeer attached to M a category C_M of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(C_M). The dual semicanonical basis contains all cluster monomials. By construction, the clu…
In this paper, we let $\Hecke$ be the Hecke algebra associated with a finite Coxeter group $W$ and with one-parameter, over the ring of scalars $\Alg=\mathbb{Z}(q, q^{-1})$. With an elementary method, we introduce a cellular basis of $\Hecke$ indexed by the sets $E_J (J\subseteq S)$ and obtain a general theory of "Specht modules". We provide an algorithm for $W\!$-graphs for the "generic Specht module", which associates with the Kazhdan and Lusztig cell ( or more generally, a union of cells of $W$ ) containing the longest element of a parabolic subgroup $W_J$ for appropriate $J\subseteq S$. As…
In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum $\mathfrak{gl}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine $\mathfrak{gl}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine $\mathfrak{gl}_n$.
For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra $\textbf{g}$ we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of $\dot{\textbf{U}}(\textbf{g})$, where the root category is the $T^2$-orbit category of the derived category of Dynkin or tame quiver. Our method bases on one theorem of Lin, Xiao and Zhang in [9], which gave th…
Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.
We establish a Schur type duality between a coideal subalgebra of the quantum group of type A and the Hecke algebra of type B with 2 parameters. We identify the $\imath$-canonical basis on the tensor product of the natural representation with Lusztig's canonical basis of the type B Hecke algebra with unequal parameters associated to a weight function.
We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M, in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincare polynomials. We also introduce a q-deformation of the Mobius algebra of M, and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra. We conjecture that t…
The complex irreducible representations of the symmetric group carry an important canonical basis called the Kazhdan-Lusztig basis. Although it is difficult to express how general permutations act on this basis, some distinguished permutations have beautiful descriptions. In 2010 Rhoades showed that the long cycle $(1, 2,..., n)$ acts by the jeu-de-taquin promotion operator in the case when the irreducible representation is indexed by a rectangular partition. We prove a generalisation of this theorem in two directions: on the one hand we lift the restriction on the shape of the partition, and …
Parallel to the very rich theory of Kazhdan-Lusztig cells in characteristic $0$, we try to build a similar theory in positive characteristic. We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see arXiv:1510.01556(2)). Our main technical tool are the star-operations introduced by Kazhdan-Lusztig which have interesting numerical consequences for the $p$-canonical basis. As an application, we explicitely describe $p$-cells in finite type $A$ (i.e. for symmetric groups) using the Robinson-Schensted correspondence. Moreover, we show th…
Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical basis $S^{*}$. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras to the dual canonical basis ${B}^{up}$. In particular, we prove that the quantum analogue $O_{q}[N(w)]$ of ${C}[N(w)]$ has the induced basis from ${B}^{up}$, which contains quantum flag minors and satisfies a factorizatio…
Let ${\mathbf U}^-_q$ be the negative part of the quantized enveloping algebra associated to a Kac-Moody algebra ${\mathfrak g}$ of symmetric type, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the orbit algebra ${\mathfrak g}^σ$ obtained from an admissible diagram automorphism $σ$ on ${\mathfrak g}$. Lusztig consructed the canonical basis ${\mathbf B}$ of ${\mathbf U}_q^-$ and the canonical signed basis $\underline{\widetilde{\mathbf B}}$ of $\underline{\mathbf U}_q^-$ by making use of the geometric theory of quivers. He proved that there is a natural bijection $\widetilde{\mat…
Let $H$ be the Iwahori-Hecke algebra and let $J$ be Lusztig's asymptotic Hecke algebra, both specialized to type $\tilde{A}_1$. For $\mathrm{SL}_2$, when the parameter $q$ is specialized to a prime power, Braverman and Kazhdan showed recently that a completion of $H$ has codimension two as a subalgebra of a completion of $J$, and described a basis for the quotient in spectral terms. In this note we write these functions explicitly in terms of the basis $\{t_w\}$ of $J$, and further invert the canonical isomorphism between the completions of $H$ and $J$, obtaining explicit formulas for the each…
We describe a positive characteristic analogue of the Kazhdan-Lusztig basis of the Hecke algebra of a crystallographic Coxeter system and investigate some of its properties. Using Soergel calculus we describe an algorithm to calculate this basis. We outline some known or expected applications in modular representation theory. We conclude by giving several examples.
