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Cohomologies on
grassmannians via derivations on
exterior algebras
Let G(k,n) be the
complex grassmannian variety
parameterizing k-planes in Cn.
Its integral singular cohomology
ring H*(G(k,n)) has been widely
investigated along the last two
centuries. Furthermore, some
geometrically relevant
"deformations" of it have been
defined and studied in the last
few decades. In his celebrated
paper "The Verlinde Algebra and
the cohomology of Grassmannians",
E. Witten introduced and begun the
study of the quantum cohomology
ring of G(k,n), which is a
suitable quantum deformation of
H*(G(k,n)). The T-equivariant
cohomology of G(k,n), instead, is
the deformation one considers when
the grassmannian comes equipped
with a certain action of a torus
T: it has been extensively studied
in an important work by Knutson
and Tao via the beautiful
combinatorics of puzzles.
The aim of the lecture will be to
show that all these different kind
of cohomology theories living on
grassmannians can be treated in a
unified way within a new, more
general and more powerful
formalism (in spite of being very
elementary) regarding derivations
of the exterior algebra of a free
module over a commutative Z-algebra.
Such a
description is also related with
some recent important work by D.
Laksov and A. Thorup, which will
be briefly discussed.
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