Interval positroid varieties and a deformation of the ring of symmetric functions - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Communication Dans Un Congrès Discrete Mathematics and Theoretical Computer Science Année : 2014

Interval positroid varieties and a deformation of the ring of symmetric functions

Résumé

Define the interval rank $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the interval rank stratification of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata interval positroid varieties. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erdős-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.
Fichier principal
Vignette du fichier
dmAT0179.pdf (422.25 Ko) Télécharger le fichier
Origine : Fichiers éditeurs autorisés sur une archive ouverte
Loading...

Dates et versions

hal-01207563 , version 1 (01-10-2015)

Identifiants

Citer

Allen Knutson, Mathias Lederer. Interval positroid varieties and a deformation of the ring of symmetric functions. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.923-934, ⟨10.46298/dmtcs.2453⟩. ⟨hal-01207563⟩

Collections

INSMI TDS-MACS
86 Consultations
693 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More