Timeline for $q$-analogs of total positivity
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2020 at 20:43 | answer | added | Jeanne Scott | timeline score: 1 | |
| Dec 13, 2019 at 13:23 | vote | accept | Christian Gaetz | ||
| Dec 13, 2019 at 0:56 | answer | added | Gjergji Zaimi | timeline score: 8 | |
| Dec 5, 2019 at 20:43 | history | edited | Sam Hopkins |
edited tags
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| Dec 5, 2019 at 18:04 | comment | added | Richard Stanley | Franceco Brenti gives a combinatorial characterization of total positivity in Theorem 4.2 of his paper in Math. Appl. 359 (1996), 451-473, repeated in JCTA 71 (1995), 175-218. Perhaps this result can be extended to $q$-total positivity or even "multivariate total positivity" to show for instance that the matrices of math.mit.edu/~rstan/papers/snf.pdf are multivariate totally positive. However, no representation theory is involved, so maybe not so interesting. | |
| Dec 5, 2019 at 17:53 | history | edited | F. C. |
added the tag q-analogs
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| Dec 5, 2019 at 16:08 | comment | added | Sam Hopkins | Also possibly related: mathoverflow.net/questions/337798/… | |
| Dec 3, 2019 at 19:42 | comment | added | Sam Hopkins | Lusztig showed that the totally positive part of $G$ is the set of $x \in G$ with $\Delta(x) \geq 0$ as $\Delta$ ranges over all elements of the dual canonical basis of the coordinate ring of $G$. Now, (roughly speaking at least) the dual canonical basis of the coordinate ring is obtained by taking a $q\to 1$ limit of the dual canonical basis for the quantized enveloping algebra $\mathcal{U}_{q}(\mathfrak{g})$. So if you could view $\mathcal{U}_{q}(\mathfrak{g})$ as "functions" on something $q$-y, you'd have a candidate for the totally positive part of that something. | |
| Dec 3, 2019 at 16:22 | comment | added | Andy Sanders | This may be irrelevant, or you may already know, but google searching cluster variety and positive structure might turn up something relevant. This paper arxiv.org/abs/math/0311149 of Fock and Goncharov is one place where the notion I have in mind turns up. | |
| Dec 3, 2019 at 16:03 | history | asked | Christian Gaetz | CC BY-SA 4.0 |