# Questions tagged [canonical-bases]

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20
questions

**24**

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**2**answers

2k 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 $...

**16**

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**2**answers

570 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 ...

**16**

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502 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 ...

**14**

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**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 ...

**11**

votes

**1**answer

656 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 $f^{(n)}$ ...

**10**

votes

**1**answer

629 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 ...

**10**

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375 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 $\...

**8**

votes

**2**answers

3k 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 ...

**8**

votes

**1**answer

835 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 $E_{12}, E_{13}, E_{23}$, $\...

**7**

votes

**1**answer

482 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 ...

**7**

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481 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 ...

**5**

votes

**0**answers

142 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 ...

**5**

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432 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 ...

**4**

votes

**1**answer

181 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 ...

**4**

votes

**1**answer

596 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 ...

**4**

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61 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 ...

**4**

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235 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 $\...

**3**

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540 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 ...

**2**

votes

**1**answer

476 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_{...

**1**

vote

**1**answer

83 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 ...