Difference between revisions of "On the boundary of Q-systems : introduction to the KNS conjecture"
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==statement of the KNS conjecture== | ==statement of the KNS conjecture== | ||
− | Let $X$ be a Dynkin diagram of type $ADE$ of rank $r$. There exists a unique solution $\mathbf{z}=(z^{(a)}_{i})$ of the level $k$ restricted $Q$-system of type $X$ satisfying $z_{i}^{(a)}>0$ for $0\leq i \leq k$ and $a\in I$. Moreover, it satisfies the following additional properties: | + | Let $X$ be a Dynkin diagram of type $ADE$ of rank $r$ and $I$ be the index set of it. There exists a unique solution $\mathbf{z}=(z^{(a)}_{i})$ of the level $k$ restricted $Q$-system of type $X$ satisfying $z_{i}^{(a)}>0$ for $0\leq i \leq k$ and $a\in I$. Moreover, it satisfies the following additional properties: |
* (symmetry) $z^{(a)}_{i}=z^{(a)}_{k-i}$ for $0\leq i \leq k$ and $a\in I$. | * (symmetry) $z^{(a)}_{i}=z^{(a)}_{k-i}$ for $0\leq i \leq k$ and $a\in I$. | ||
* (unimodality) $z^{(a)}_{i-1}<z^{(a)}_{i}$ for $1\le i \le m=\lfloor\frac{k}{2}\rfloor$ and $a\in I$ where $\lfloor x\rfloor$ is the floor function. | * (unimodality) $z^{(a)}_{i-1}<z^{(a)}_{i}$ for $1\le i \le m=\lfloor\frac{k}{2}\rfloor$ and $a\in I$ where $\lfloor x\rfloor$ is the floor function. |
Revision as of 17:57, 26 December 2012
talks
- Séminaire d'Algèbre, IHP, 3/12/2012
statement of the KNS conjecture
Let $X$ be a Dynkin diagram of type $ADE$ of rank $r$ and $I$ be the index set of it. There exists a unique solution $\mathbf{z}=(z^{(a)}_{i})$ of the level $k$ restricted $Q$-system of type $X$ satisfying $z_{i}^{(a)}>0$ for $0\leq i \leq k$ and $a\in I$. Moreover, it satisfies the following additional properties:
- (symmetry) $z^{(a)}_{i}=z^{(a)}_{k-i}$ for $0\leq i \leq k$ and $a\in I$.
- (unimodality) $z^{(a)}_{i-1}<z^{(a)}_{i}$ for $1\le i \le m=\lfloor\frac{k}{2}\rfloor$ and $a\in I$ where $\lfloor x\rfloor$ is the floor function.
- $z^{(a)}_{k+1}=z^{(a)}_{k+2}=\cdots =z^{(a)}_{k+h-1}=0$ for $a\in I$.
example : level $k=8$ restricted Q-system of type $D_5$
$$ \begin{bmatrix}
1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 6.027 & 19.16 & 42.36 & 8.524 & 8.524 \\ 17.16 & 112.0 & 401.6 & 30.30 & 30.30 \\ 30.30 & 294.6 & 1380. & 60.60 & 60.60 \\ 36.33 & 401.6 & 2050. & 75.65 & 75.65 \\ 30.30 & 294.6 & 1380. & 60.60 & 60.60 \\ 17.16 & 112.0 & 401.6 & 30.30 & 30.30 \\ 6.027 & 19.16 & 42.36 & 8.524 & 8.524 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0
\end{bmatrix} $$
- introduction to the KNS conjecture about quantum dimension solution of Q-systems
- contains explicit examples of level $k$ restricted Q-systems
- File:IntroKNS.pdf