MARM--2020-2021-Project HiDEA

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The London Mathematical Society Program on

MENTORING AFRICAN RESEARCHERS in MATHEMATICS

at the University of NAMIBIA
Academic Year 2020-2021

(grant holder: Letterio Gatto -- UNAm Marm referent: Martin Mugochi)

Research Project HiDEA
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The aim of the project is to investigate the interdisciplinary connections among the algebra, geometry, physics and combinatorics of Higher Derivations on Exterior Algebras.

HiDEA Seminar on

Multiplying exterior powers by matrices

by Ommolbanin Behzad

University of Isfahan

August 3, 2021, 2:00 pm (Windhoek time)

access via the zoom link:

https://us02web.zoom.us/j/81783056071?pwd=RmtDMHpjSEZwUlR2VHVWZVpHc1BIQT09

Abstract. The easiest example of an exterior power of a vector space is the vector space itself, corresponding to the first degree of its exterior algebra. We will show how an exterior power of Kⁿ is a module over the Lie algebra of the K^n×n matrices. To this purpose, we will compute a generating function encoding for the usual multiplication of matrices by vectors and we shall indicated how to perform the same kind of computation for higher exterior powers by using the notion of Schubert derivation. The talk requires a minimum of prerequisites from multilinear algebra.

	Meetings' Schedule
	Forthcoming meetings will be scheduled soon

Past 2020 Meetings

Provisional Activity Plan
(in progress)

Will we plan any
Question Time?

Meeting 005, 2020, September 30,

Exterior Algebras, Exterior Powers and Partitions

Synopsis: Partitions parametrise bases of the exterior powers. Young Diagrams of partitions

Notes

Meeting 004, 2020, September 23,

Exterior Algebras associated to a Vector Space

Synopsis: The construction and the definition of the esterior algebra supported on the direct sum the exterior powers

Notes

Meeting 003, 2020, September 09

Towards Exterior Algebras through Exterior Powers

Synopsis: A quick recall on what a derivation of an R-algebra is. Exterior powers of a vector space are introduced through examples and exercises to get familiar with the notion.

Notes

Meeting 002, 2020, September 02,

Associative Algebras, Lie Algebras. Definitions and Examples.

Synopsis: A quick revision of the notion of algebras over a commutative ring.

Exercise List 002

Notes

Meeting 001, 2020, August 25,

An introduction to Hasse-Schmidt Derivations on algebras in general and proposal/description of possible research directions (Notes)

Some References
books | papers
(to download)

Books

Schubert calculus: an algebraic introduction,

Publicações Matemáticas do IMPA, [IMPA Mathematical Publications] 25^o Colóquio Brasileiro de Matemática, [25th Brazilian Mathematics Colloquium] Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2005

Linear ODEs: an Algebraic Perspective,

XXII Escola de Algebra, Salvador de Bahia, 2012, Publicações Matemáticas do IMPA, n. 40, 2012.

Generic Linear Recurrent Sequences and Related Topics,

30° Colóquio Brasileiro de Matemática, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 26‐31 de Julho 2016

Papers

Polynomial Ring Representations of Endomorphisms of Exterior Powers, Collectanea Math. 2021, ArXiv:2005.01154v2,

Schubert Derivations on the Infinite Wedge Powers, Bulletin of the Brazilian Mathematical Society, New Series, Springer, 2020 (with P. Salehyan)

The Cohomology of the Grassmannian is a gl_n-module, Comm. Algebra 2020, 48, no. 1, 274–290

Hasse–Schmidt Derivations and Cayley–Hamilton Theorem for Exterior Algebras, Contemporary Mathematics 733, AMS, 2019 (with I. Scherbak)

Linear ODE's, Wronskian and Schubert Calculus, Moscow Mat. J., 12, no. 2, Special issue in memory of V. I. Arnold, (2012) 275–291 (with I. Scherbak);

Schubert Calculus via Hasse-Schmidt derivations, Asian J. Math. 9 (2005), no. 3. 315-321