{
"id": "1812.08345",
"version": "v1",
"published": "2018-12-20T03:14:45.000Z",
"updated": "2018-12-20T03:14:45.000Z",
"title": "Minuscule reverse plane partitions via quiver representations",
"authors": [
"Alexander Garver",
"Rebecca Patrias",
"Hugh Thomas"
],
"comment": "Comments welcome",
"categories": [
"math.RT",
"math.CO"
],
"abstract": "A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If $Q$ is a Dynkin quiver and $m$ is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including $m$ in their support, the category of which we denote by $\\mathcal C_{Q,m}$, are determined up to isomorphism by this invariant. We use this invariant to define a bijection from isomorphism classes of representations in $\\mathcal C_{Q,m}$ to reverse plane partitions whose shape is the minuscule poset corresponding to $Q$ and $m$. By relating the piecewise-linear promotion action on reverse plane partitions to Auslander-Reiten translation in the derived category, we give a uniform proof that the order of promotion equals the Coxeter number. In type $A_n$, we show that special cases of our bijection include the Robinson-Schensted-Knuth and Hillman-Grassl correspondences.",
"revisions": [
{
"version": "v1",
"updated": "2018-12-20T03:14:45.000Z"
}
],
"analyses": {
"subjects": [
"16G20",
"05E10"
],
"keywords": [
"minuscule reverse plane partitions",
"linear transformation",
"nilpotent endomorphism",
"quiver representation induces",
"piecewise-linear promotion action"
],
"note": {
"typesetting": "TeX",
"pages": 0,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}