Hi,

The following formulas are examples of non-positive structure coefficients for non-symmetric cases which are easily verified by the algorithm presented in Leclerc's paper "Dual Canonical Bases, Quantum Shuffles, and q-characters" or quagroup package in GAP4.

Professor Masaki Kashiwara told me that he has known such non-positive structure coefficient for $G_2$ since Shigenori Yamane found it in 1994 as treated in his master thesis at Osaka University (written in Japanese). You can see similar negative coefficients in at least case $A_{2n}^{(2)}, D_{n+1}^{(2)}$. Anyway, conjecture 52 in Leclerc's paper is false (I already told Professor Leclerc about it).

Shunsuke Tsuchioka

Notation: $G(i_1,\cdots,i_n)$ stands for the canonical basis element corresponds to a crystal element $b(i_1,\cdots,i_n)=\tilde{f}_{i_n}b(i_1,\cdots,i_{n-1})=\cdots$.

$G_2$ (1 is the short root) : $f_2 G(121112211) = G(1211122211) + [2]G(1111222211) + G(2111112221) G(2111112221)$

$+ [2]G(1211112221) + G(1111122221) - G(1112211122) + [2]G(1122111122)$

$C_3$ (1,2 are short roots) : $f_3 G(23122312) = [2]G(222333121) + [2]G(312222331) + [2]G(231222331) 2]G(231222331)$

$+ [2]G(122223331) + G(231223312) + [2]G(122233312) - G(223112233) + [2]G(231122233)$

$B_4$ (1,2,3 are long roots) : $f_1G(4342341234) = [2]G(43344423211) + [2]G(43423443211) - G(44233443211) G(44233443211)$

$+ [2]G(43423344211) + [2]G(43423442311) + [2]G(34234442311) 2]G(34234442311)$

$+ [2]G(43422334411) + G(43423412341)$

1

Hi,

The following formulas are examples of non-positive structure coefficients for non-symmetric cases which are easily verified by the algorithm presented in Leclerc's paper "Dual Canonical Bases, Quantum Shuffles, and q-characters" or quagroup package in GAP4.

Professor Masaki Kashiwara told me that he has known such non-positive structure coefficient for $G_2$ since Shigenori Yamane found it in 1994 as treated in his master thesis at Osaka University (written in Japanese). You can see similar negative coefficients in at least case $A_{2n}^{(2)}, D_{n+1}^{(2)}$. Anyway, conjecture 52 in Leclerc's paper is false (I already told Professor Leclerc about it).

Shunsuke Tsuchioka

Notation: $G(i_1,\cdots,i_n)$ stands for the canonical basis element corresponds to a crystal element $b(i_1,\cdots,i_n)=\tilde{f}_{i_n}b(i_1,\cdots,i_{n-1})=\cdots$.

$G_2$ (1 is the short root) : $f_2 G(121112211) = G(1211122211) + [2]G(1111222211) + G(2111112221) + [2]G(1211112221) + G(1111122221) - G(1112211122) + [2]G(1122111122)$

$C_3$ (1,2 are short roots) : $f_3 G(23122312) = [2]G(222333121) + [2]G(312222331) + [2]G(231222331) + [2]G(122223331) + G(231223312) + [2]G(122233312) - G(223112233) + [2]G(231122233)$

$B_4$ (1,2,3 are long roots) : $f_1G(4342341234) = [2]G(43344423211) + [2]G(43423443211) - G(44233443211) + [2]G(43423344211) + [2]G(43423442311) + [2]G(34234442311) + [2]G(43422334411) + G(43423412341)$