23.06.26 (Milano and some Catalan fun-fact)
I was for the first time math-wise in Italy (in Milano where I went to GABY 2026). It was so cool and I was really convinced by the underground university architecture to avoid heat.
The following fun-fact might brighten up your mood even in the darkest moments (If you like Catalan numbers(Wikipedia)). Here it comes: The determinant of the matrix which puts Catalan numbers on main diagonals like (1 & 1 \\ 1 & 2) or (1 & 1 & 2 \\ 1 & 2 & 5 \\ 2 & 5 & 14) is 1. Moreover this matrix is symmetric. Even better it factors as a product of a matrix and it's transpose.
For instance (1 & 1 & 2 \\ 1 & 2 & 5 \\ 2 & 5 & 14) = A * A_transposed, where A=(1 & 0 & 0 \\ 1 & 1 & 0 \\ 2 & 3 & 1)).
Of course this has to have deep algebraic meaning, such nice stuff doesn't happen without a reason. The fun part: The entries of A that are written there are the dimensions of simple representations of Temperley-Lieb algebras (or the symmetric groups corresponding to partitions with at most 2 rows). This sort of factorization of integer matrices in a standard phenomenon of Cartan matrices for quasi-hereditary algebras. So you might ask what is the algebra here? Well it is the Temperley--Lieb category considered as a path algebra modulo relations. For instance (just take the even natural numbers), take 3 vertices called 0, 2, and 4. Draw 1 arrow from 0 to 2 and back, draw 3 arrows from 2 to 4 and back and 5 arrows from 4 to 6 and back (corresponing to certain cup Temperley-Lieb diagrams) and put the according diagrammatic relations.
What is still confusing for me: The difference between the representation theory of the Temperley-Lieb algebras and this actual TL category as quiver. For the Temperley-Lieb category as a quiver as described here the simples are 1-dimensional and the standard representations are built of out the simple representations of the Temperley--Lieb algebras (for instance for TL_4 we have a 2, a 3 dimensional and a 1-dimensional representation). What is the meaning of the Koszul dual? The numbers in the inverse are certain binomial coefficients and explain how to decompose 2-color Soergel bimodules for dihedral type (read dihedral cathedral or the survey by Elias and Williamson on the diagrammatics for Soergel bimodules).