Questions tagged [canonical-bases]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2 votes
0 answers
114 views

Canonical basis and perverse coherent sheaves on the nilpotent cone

In the paper of Ostrik, he introduced a canonical basis of $K^{G\times {\mathbb C}^*}(\mathcal N)$, where $\mathcal N$ is the nilpotent cone for the group $G$. Question: does this canonical basis ...
Yellow Pig's user avatar
  • 2,480
3 votes
0 answers
76 views

Theta functions in acyclic cluster algebras

Setup: Let $B$ be a skew-symmetrisable integer matrix and $\mathcal{A}$ be the cluster algebra with principal coefficients at a seed with mutation matrix $\tilde{B}$ whose principal part is equal to $...
Antoine de Saint Germain's user avatar
2 votes
0 answers
82 views

Canonical basis of the invariant part of $O_q(\mathfrak g)^{\otimes N}$

Let $\mathfrak g$ be a semi-simple Lie algebra (We can assume $\mathfrak g=sl(n)$ for simplicity) and let $O_q(\mathfrak g)$ be the corresponding quantum algebra of functions. Then $O_q(\mathfrak g)^{\...
Adam's user avatar
  • 2,370
4 votes
1 answer
203 views

Schauder basis of the Hardy space of semi-martingales

Fix $p\in [1,2]$, a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$, and let $\mathcal{H}_{\mathscr{S}}^p$ denote the space of semimartingales $X$ such that the norm $$ \...
Carlos_Petterson's user avatar
19 votes
2 answers
1k views

Is it possible that the GHKK canonical basis for cluster algebras is the Lusztig/Kashiwara dual canonical basis?

Gross-Hacking-Keel-Kontsevich (https://arxiv.org/abs/1411.1394) constructed a canonical basis (the so-called “theta basis”) for a cluster algebra, at least assuming it satisfies a certain ...
Sam Hopkins's user avatar
  • 22.5k
4 votes
0 answers
74 views

Do global bases exist for quantum enveloping algebras at $q$ nonroot of unity?

Take $\Bbbk$ to be a field, $q \in \Bbbk$ a nonroot of unity, and $U = U_q(\mathfrak g)$ the quantized enveloping algebra of a complex finite dimensional simple Lie algebra, and write $U^-$ for its ...
Pablo Zadunaisky's user avatar
6 votes
0 answers
195 views

Combinatorics of $p$-Kazhdan--lusztig polynomials

When can we (and can we not!) understand the dimensions of simple modules, $D(\lambda)$, of symmetric groups in a combinatorial fashion? Let's assume that I'm going to try to do this using the theory ...
Chris Bowman's user avatar
  • 1,191
1 vote
1 answer
107 views

Question about a computation of $F_{\beta_3}$ related to canonical basis of $U_q(sl_3)$.

I am reading the paper: ELEMENTARY CONSTRUCTION OF LUSZTIG’S CANONICAL BASIS and want to compute $F_{\beta_3}$ in Example 3 on page 3. Maybe there is some mistake in my computations but I could not ...
Jianrong Li's user avatar
  • 6,101
4 votes
1 answer
238 views

Bases closed under multiplication

Let us say that a Hamel basis $H$ in an algebra $A$ is closed under multiplication, if $ab\in H$ whenever $a,b\in H$. It is an easy observation that if $A$ has such a basis then there it also has a ...
Tomasz Kania's user avatar
  • 11.3k
8 votes
1 answer
1k views

PBW basis and canonical basis

Consider the example of $\mathfrak{g} = sl_3$. Then $$ \mathfrak{g} = \mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^{-}, $$ where $\mathfrak{n}$ is generated by $E_{12}, E_{13}, E_{23}$, $\...
Jianrong Li's user avatar
  • 6,101
3 votes
0 answers
797 views

Canonical basis of quantum groups

I am trying to understand the canonical basis of quantum groups and different ways to construct the canonical basis of quantum groups. In the comments of Lusztig's papers, the paper [92], CANONICAL ...
Jianrong Li's user avatar
  • 6,101
11 votes
2 answers
6k views

Canonic identification of the tangent space of the Grassmannian

let $Gr(k,V)$ be the grassmannian of k-dimensional subspaces of the complex vector space $V$ of dimension $n>k$. I know that, given $K\in Gr(k,V)$, $T_{Gr(k,V),K}\simeq Hom(K,V/K)$, but i want to ...
Tom Fellmann's user avatar
14 votes
2 answers
1k views

Where does the canonical basis differ from the KLR basis?

The question implicitly asked in Ben Webster's question is: Does the canonical basis of Uq(n+) agree with the basis coming from categorification via Khovanov-Lauda-Rouqier algebras? Thanks to ...
Peter McNamara's user avatar
5 votes
0 answers
474 views

Are Lusztig's perverse sheaves the only equivariant ones with nilpotent characteristic cycle?

In his '91 paper, Lusztig defines a collection of simple perverse sheaves that correspond to the canonical basis; these are defined using a pushforward construction, and from the definition, it's easy ...
Ben Webster's user avatar
  • 43.9k
10 votes
0 answers
513 views

Explicit change of basis for the Schur-Weyl basis

The Schur-Weyl duality states that $\bigotimes_{m=1}^n \mathbb{C}^k$ can be decomposed as a direct sum over the tensor product of irreductible representations of $SU(k)$ and of the symmetric group $\...
Damien S.'s user avatar
  • 244
7 votes
0 answers
516 views

Where can I find tables of dual canonical basis vectors?

Leclerc (arXiv:math/0209133) has given us an algorithm for computing the dual canonical basis of the upper part of a quantised enveloping algebra. Now presumably this algorithm has been implemented ...
Peter McNamara's user avatar
3 votes
1 answer
528 views

What is the current status for Lusztig's positivity conjecture for symmetric Cartan datum?

This is related to the earlier question here In Conjecture 25.4.2 in his "Introduction to Quantum Groups," Lusztig conjectures that "If the Cartan datum is symmetric, then the structure constants $m_{...
Marty's user avatar
  • 13.1k
16 votes
0 answers
525 views

Comparing the Kazhdan-Lusztig and Steinberg pre-orders

Both Kazhdan-Lusztig and Steinberg have defined pairs of preorders on $S_n$. Kazhdan and Lusztig's preorders come from their basis: We write $x\leq_L y$ if any left ideal spanned by K-L basis ...
Ben Webster's user avatar
  • 43.9k
26 votes
2 answers
3k views

When does Lusztig's canonical basis have non-positive structure coefficients?

I've heard asserted in talks quite a few times that Lusztig's canonical basis for irreducible representations is known to not always have positive structure coefficents for the action of $E_i$ and $...
Ben Webster's user avatar
  • 43.9k
4 votes
0 answers
241 views

Analogy between canonical basis of U(n_-) and Schur functors, each under restriction

.1. For any category $\mathcal C$, possibly enriched over schemes, define $Rep({\mathcal C})$ to be the functor category ${\mathcal C} \to {\bf Vec}$ with direct sum inherited from $\bf Vec$. (If $\...
Allen Knutson's user avatar
11 votes
1 answer
773 views

What's known about the stalks of Lusztig's perverse sheaves on quiver varieties?

Lusztig has defined a category of perverse sheaves on the moduli space of representations of a Dynkin quiver (see his paper) corresponding to canonical basis vectors. I'm interested in the stalks ...
Ben Webster's user avatar
  • 43.9k
12 votes
1 answer
868 views

Do Jones-Wenzl idempotents lift to anything interesting in the Hecke algebra?

Background Inside the Temperley-Lieb algebra $TL_n$ (with loop value $\delta=-[2]$ and standard generators $e_1,\ldots,e_{n-1}$), the Jones-Wenzl idempotent is the unique non-zero element $f^{(n)}$ ...
Sammy Black's user avatar
  • 1,746
4 votes
1 answer
704 views

Canonical basis for the extended quantum enveloping algebras

I am trying to understand some construction done by Lusztig in his book on quantum groups. Given some Cartan datum, let $U=U_q(\mathfrak{g})$ the standard quantized enveloping algebra of the Kac-Moody ...
javier's user avatar
  • 2,911
7 votes
1 answer
493 views

Does some version of U_q(gl(1|1)) have a basis like Lusztig's basis for \dot{U(sl_2)}?

There's a non-unital algebra $\dot{U}$ formed from $U_q (sl_2)$ by including a system of mutually orthogonal idempotents $1_n$, indexed by the weight lattice. You can think of this as a category with ...
Sammy Black's user avatar
  • 1,746