# Big cells in a Grassmann and permutations

In the lecture notes, it is said that (Theorem 3.1.3) the set of positroid cells in $Gr(k,n)$ are in one to one correspondence with the set of bounded affine permutations of type $(k,n)$. In Example 4.1.5, it is said that the permutation $\sigma$ corresponds to the big cell $Gr(3,6)$. What are the permutations which corresponds to non-big cells? Thank you very much.

• @Sam, editing 16 old questions in the space of a few hours drives 16 new questions off the front page. Please don't do that. – Gerry Myerson Dec 6 '19 at 2:41
• @GerryMyerson: apologies, I was trying to populate a new tag, I didn't realize it would have that effect. – Sam Hopkins Dec 6 '19 at 2:43

Positroid cells in $Gr(k,n)$ are indexed by many objects we often want to go between. The big cell will be given by the bounded affine permutation $i \mapsto i+k$. See Postnikov's original preprint (section 16). Note the positroid for the big cell will the the positroid consisting of all $k$ subsets of $[n]$. In the preprint decorated permutations are used, bounded affine permutations are the language of Knutson-Lam-Speyer. Section 16 of the preprint also makes use of Grassmann necklaces which are another object indexing positroid cells. Any other decorated permutation/bounded affine permutation of type $(k,n)$ gives a non-big cell.