Equating decomposition numbers for different primes

A Prime Decomposition of Probabilistic Automata

A definition of a probabilistic automaton is formulated in which its prime decomposition follows as a direct consequence of Krohn-Rhodes theorem. We first characterize the local structure of probabilistic automata. The prime decomposition is presented as a framework to study the global structure of probabilistic automata. We prove that the representation theory of a probabilistic automaton is determined by that of the finite groups in its holonomy decomposition.

Hernandez-Leclerc范畴中的素表示：经典分解

Prime representations in the Hernandez-Leclerc category: classical decompositions

本文研究量子仿射代数HL范畴中的素不可约表示，确定其经典分解与分次极限重数，进一步得到仿射李代数二级可积表示中稳定素Demazure模的分次分解。

On the Decomposition Numbers of the Ree Groups 2F4(q^2) in Non-Defining Characteristic

We compute the l-modular decomposition matrices of the simple Ree groups 2F4(q^2), where q^2=2^{2n+1} and n is a positive integer, for all primes l > 3 up to some entries in the unipotent characters. Using these matrices we determine the smallest degree of a non-trivial irreducible l-modular representation of 2F4(q^2) for all primes l > 3. We also obtain results on the 3-modular decomposition matrices of 2F4(q^2).

Decomposition numbers of SO_7(q) and Sp_6(q)

We complete the l-modular decomposition numbers of the unipotent characters in the principal block of the special orthogonal groups SO_7(q) and the symplectic groups Sp_6(q) for all prime powers q and all odd primes l different from the defining characteristic.

On refined Bruhat decompositions and endomorphism algebras of Gelfand-Graev representations

Let $G$ be a finite reductive group defined over $\mathbb{F}_q$, with $q$ a power of a prime $p$. Motivated by a problem recently posed by C. Curtis, we first develop an algorithm to express each element of $G$ into a canonical form in terms of a refinement of a Bruhat decomposition, and we then use the output of the algorithm to explicitly determine the structure constants of the endomorphism algebra of a Gelfand-Graev representation of $G$ when $G=\mathrm{PGL}_3(q)$ for an arbitrary prime $p$, and when $G=\mathrm{SO}_5(q)$ for $p$ odd.

Decomposition numbers for the principal $Φ_{2n}$-block of $\mathrm{Sp}_{4n}(q)$ and $\mathrm{SO}_{4n+1}(q)$

We compute the decomposition numbers of the unipotent characters lying in the principal $\ell$-block of a finite group of Lie type $B_{2n}(q)$ or $C_{2n}(q)$ when $q$ is an odd prime power and $\ell$ is an odd prime number such that the order of $q$ mod $\ell$ is $2n$. Along the way, we extend to these finite groups the results of \cite{DVV19} on the branching graph for Harish-Chandra induction and restriction.