All Questions
Tagged with quantum-groups canonical-bases
9 questions
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 $...
- 44.7k
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 ...
- 8,866
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 $E_{12}, E_{13}, E_{23}$, $\...
- 6,201
7
votes
0
answers
528
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 ...
- 8,866
4
votes
1
answer
731
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 ...
- 2,941
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 ...
- 651
3
votes
0
answers
869
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 ...
- 6,201
2
votes
0
answers
84
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)^{\...
- 2,390
1
vote
1
answer
110
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 ...
- 6,201