On Mirković-Vilonen cycles and crystals combinatorics
| dc.creator | Baumann, Pierre | |
| dc.creator | Gaussent, Stéphane | |
| dc.date | 2006-06-28 | |
| dc.date | 2008-04-24 | |
| dc.date.accessioned | 2026-07-07T09:34:32Z | |
| dc.date.available | 2026-07-07T09:34:32Z | |
| dc.description | Let $G$ be a complex reductive group and let $G^\vee$ be its Langlands dual. Let us choose a triangular decomposition $\mathfrak g^\vee=\mathfrak n^\vee_-\oplus\mathfrak h^\vee\oplus\mathfrak n^\vee_+$ of the Lie algebra $G^\vee$. Braverman, Finkelberg and Gaitsgory show that the set of all Mirković-Vilonen cycles in the affine grassmannian $\mathscr G=G\bigl(\mathbb C((t))\bigr)/G\bigl(\mathbb C[[t]]\bigr)$ is a crystal isomorphic to the crystal of the canonical basis of $U(\mathfrak n^\vee_+)$. Starting from the string parameter of an element of the canonical basis, we give an explicit description of a dense subset of the associated MV cycle. As a corollary, we show that any MV cycle can be obtained as the closure of one of the varieties involved in Lusztig's algebraic-geometric parametrization of the canonical basis. In addition, we prove that the bijection between LS paths and MV cycles constructed by Gaussent and Littelmann is an isomorphism of crystals. | |
| dc.description | This is the very new version | |
| dc.identifier | https://arxiv.org/abs/math/0606711 | |
| dc.identifier | http://arxiv.org/abs/math/0606711 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/159527 | |
| dc.subject | Representation Theory | |
| dc.subject | 20G05 | |
| dc.title | On Mirković-Vilonen cycles and crystals combinatorics | |
| dc.type | text |