When does Lusztig's canonical basis have non-positive structure coefficients? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T14:13:20Z http://mathoverflow.net/feeds/question/39934 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39934/when-does-lusztigs-canonical-basis-have-non-positive-structure-coefficients When does Lusztig's canonical basis have non-positive structure coefficients? Ben Webster 2010-09-25T06:25:38Z 2010-10-16T00:06:35Z <p>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 $F_i$. There are good geometric reasons the coefficents have to be positive in simply-laced situations, but no such arguments can work for non-simply laced examples. However, this is quite a bit weaker than knowing the result is false.</p> <blockquote> <p>Does anyone have a good example or reference for a situation where this positivity fails?</p> </blockquote> http://mathoverflow.net/questions/39934/when-does-lusztigs-canonical-basis-have-non-positive-structure-coefficients/40556#40556 Answer by David Hill for When does Lusztig's canonical basis have non-positive structure coefficients? David Hill 2010-09-29T23:44:21Z 2010-10-16T00:06:35Z <p>Ben,</p> <p>I have heard the same thing, but I have never seen an example. After thinking about it a bit, I came up with the following 'heuristic' reason why the structure constants should be positive for half the Chevalley generators.</p> <p>Assume that it is know that simple modules for affine quiver Hecke algebras have characters given by the dual canonical basis of $U_q({n})$ (many people (not including Nakajima) expected to be true, but in light of Tsuchioka's answer shouldn't!). References for what follows are Leclerc's paper <a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=TI&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=leclerc&amp;s5=&amp;s6=&amp;s7=shuffle&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=1&amp;mx-pid=2045836" rel="nofollow">"Dual Canonical Bases, Quantum Shuffles, and $q$-characters"</a> (<strong>EDIT</strong> (BW) This is also on the <a href="http://arxiv.org/abs/math/0209133" rel="nofollow">arXiv</a>) as well as the paper of <a href="http://front.math.ucdavis.edu/0909.1984" rel="nofollow">Kleshchev and Ram</a>, and <a href="http://front.math.ucdavis.edu/0912.2067" rel="nofollow">my paper with Melvin and Mondragon</a>.</p> <p>We denote the canonical and dual canonical bases $b_g$ and $b_g^*$, respectively, where $g$ runs over an appropriate index set. These bases are related by the Kashiwara form $(\cdot,\cdot)_K:U_A(n)\times U_A^*(n)\to A$ via </p> <p>$$(b_g,b_h^*)<em>K=\delta</em>{gh}$$</p> <p>(above, $A=\mathbb{Z}[q,q^{-1}]$ as usual). This form is defined so that $(1,1)_K=1$ and $(f_iu,v)_K=(u,f_i'v)_K$ and $f_i'$ is Kashiwara's $q$-derivation.</p> <p>Now, on the level of modules, the $q$-derivation $f_i'$ corresponds to $i$-restriction. As we have assumed $b^*_g$ is the character of a simple module, we have the $f_i'\mathcal{b}^*_g$ is the character of some module, and hence a nonnegative linear combination of dual canonical basis vectors. So now we calculate \begin{align*} f_i\mathcal{b}_g=\sum_h(f_ib_g,b^*_h)_Kb_h =\sum_h(b_g,f_i'b^*_h)_Kb_h. \end{align*} But, as we have explained, $(b_g,f_i'b^*_h)$ is nonnegative.</p> <p>As I said above, this argument only works for half the generators. I haven't internalized the results of your recent paper, so I'm not sure if you've defined the biadjoint functor $e_i$ or not. If you have, then probably there should be some more information to be teased out of this line of reasoning.</p> http://mathoverflow.net/questions/39934/when-does-lusztigs-canonical-basis-have-non-positive-structure-coefficients/40577#40577 Answer by Shunsuke Tsuchioka for When does Lusztig's canonical basis have non-positive structure coefficients? Shunsuke Tsuchioka 2010-09-30T05:29:33Z 2010-09-30T08:35:35Z <p>Hi, </p> <p>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.</p> <p>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).</p> <p>Shunsuke Tsuchioka</p> <p>Notation: $G(i_1,\cdots,i_n)$ stands for the canonical basis element corresponds to a crystal element $b(i_1,\cdots,i_n)=\tilde{f}_{i_n}b(i_1,\cdots,i_{n-1})=\cdots$. </p> <p>$G_2$ (1 is the short root) : $f_2 G(121112211) = G(1211122211) + [2]G(1111222211) + G(2111112221)$</p> <p>$+ [2]G(1211112221) + G(1111122221) - G(1112211122) + [2]G(1122111122)$</p> <p>$C_3$ (1,2 are short roots) : $f_3 G(23122312) = [2]G(222333121) + [2]G(312222331) + [2]G(231222331)$</p> <p>$+ [2]G(122223331) + G(231223312) + [2]G(122233312) - G(223112233) + [2]G(231122233)$</p> <p>$B_4$ (1,2,3 are long roots) : $f_1G(4342341234) = [2]G(43344423211) + [2]G(43423443211) - G(44233443211)$</p> <p>$+ [2]G(43423344211) + [2]G(43423442311) + [2]G(34234442311)$</p> <p>$+ [2]G(43422334411) + G(43423412341)$</p>