Summer 2024

15.08.24 (Summer, Nil-Hecke, being careful with notation 1 for the identity matrix, even and odd polys)

Hello everyone, from 15.07-19.07 I was in Leipzig, on this summer school on Algebraic Combinatorics, followed by FPSAC Bochum 2024 the big Algebraic Combinatorics conference (22.07.-26.07), which was for the first time held in Germany. It was very nice, to meet so many new people from far away places. Many talks were interested for an algebra/rep theory enjoyer like me. Currently it's wayyyyyy to hot in Dresden.

Anyways let's go back to the present and discuss some ideas/thoughts on my current Nil--Hecke project, which I try to write in such a way that it adresses two communities (In the paper one can actually understand something, unlike this mediocre blog post.) However technically speaking it is known to both communities (at least it appeared in both areas however utilizing different languages). 

The first thing, was that Myriam Mahanam explained to me that it is known that End_R(A) is an A-bialgebroid given that B\subset A are two commutative rings, such that A is finite free as B-module (Think A=k[x], B=k[x^2]=A^{S_2}). Now I finally understood how this works by reading Sweedlers 'Groups of Simple algebras' more carefully. Namely A*=Hom_B(A,B) becomes a B-coalgebra, extension of scalars turns A\otimes_B A^{*}= End_B(A) into an A-coalgebra, and note here that A\otimes_B A^{*} \otimes_B A^{*} = A\otimes_B A^{*} \otimes_A A\otimes_B A^{*}=End_B(A) \otimes_A End_B(A).

One can show that the resulting comultiplication defines an iso on the Takeuchi-product End_B(A) \times_A End_B(A), which is an algebra morphism and hence a bialgebroid.

Back to reality. What happens here is that the basis 1, x of A=k[x] as B=k[x]/x^2 module gives (k[x]^*) has then dual basis 1*, x*) and one has the B-basis of End_B(A) given by 1\otimes 1*, 1\otimes x*, x\otimes 1, x\otimes x* and 1 of this algebra is 1\otimes 1* +  x\otimes x*. One can check that 2 \cdot 1\otimes x^*=\del_s is the Demazure/divided difference operator in this language and that the comultiplication becomes \del_s \otimes s + 1\otimes \del_s.