The 0-Rook, and 0-Renner Monoids
Time: 11:00 -- Location: Salle Philippe Flajolet du LIX
We show that a proper degeneracy at \(q = 0\) of the \(q\)-deformed rook monoid of Solomon is the algebra of a monoid \(R_n^0\) namely the 0-rook monoid, in the same vein as Norton's 0-Hecke algebra being the algebra of a monoid \(H_n^0 := H_n^0(A)\) (in Cartan type A). As expected, \(R_n^0\) is closely related to the latter: it contains the \(H_n^0(A)\) monoid and is a quotient of \(H_n^0(B)\). This talk will be about the way to define it, using actions on permutations and rook vectors, with some algorithms.
Finally, the rook monoid is a particular case of a family of monoids indexed by the Weyl groups, called the Renner monoids. We show that we can also define a "0-Renner monoid", and use this to give a proper characterization of elements of renner monoids.