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A fairly old problem in modular representation theory is to determine the vanishing behavior of the $H o m$ groups and higher $E x t$ groups of Weyl modules and to compute the dimension of the $ℤ / (p)$ -vector space ${H o m}_{{\bar{A}}_{r}} ({\bar{K}}_{λ}, {\bar{K}}_{μ})$ for any partitions $λ$ , $μ$ of $r$ , which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups ${H o m}_{{\bar{A}}_{r}} ({\bar{K}}_{λ}, {\bar{K}}_{μ})$ and provide a new formula for the intertwining number for any...

Invariants, torsion indices and oriented cohomology of complete flags

Baptiste Calmès, Viktor Petrov, Kirill Zainoulline (2013)

Annales scientifiques de l'École Normale Supérieure

Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$ . Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$ -theory, Chow groups, Grothendieck’s $K_{0}$ , etc.) with formal group law $F$ . We construct a ring from $F$ and the characters of $T$ , that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁 (G / B)$ where $G / B$ is the variety of Borel subgroups of $G$ . Our main result says that when the torsion index...

Isomorphisms in the Farrell cohomology of $Sp (p - 1, ℤ [1 / n])$ .

Busch, Cornelia Minette (2008)

The New York Journal of Mathematics [electronic only]

$L_{2}$ -cohomology and intersection homology of locally symmetric varieties, II

Steven Zucker (1986)

Compositio Mathematica

L2 Cohomology and Warped Products and Arithmetic Groups.

Steven Zucker (1982/1983)

Inventiones mathematicae

Linear free divisors and the global logarithmic comparison theorem

Michel Granger, David Mond, Alicia Nieto-Reyes, Mathias Schulze (2009)

Annales de l’institut Fourier

A complex hypersurface $D$ in $ℂ^{n}$ is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for $n$ at most $4$ .By analogy with Grothendieck’s comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for $D$ if the complex of global logarithmic differential forms computes the complex cohomology of $ℂ^{n} ∖ D$ . We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the...