Schurweyl duality consider the representations induced by these commuting actions, cs k. The centralizer of gl n acting on tensor space m b v bk b n displays type c combinatorics for good choices of m, n,andv. Applications of frobenius algebras to representation. Rational cherednik algebras, quiver schur algebras and. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum eld theory. Schur at the begining of the 20th century, is a powerful tool to study polynomial representations of general linear group. Stuart martin the schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups both finite and infinite. The connection is based on a remarkable homological property discovered in dps enjoyed by cell filtrations of certain induced or q permutation modules for hecke algebras.
They are part of my phd thesis schur algebras of classical groups written in chinese, defended in may. In the group case it says that if m and n are two finitedimensional irreducible representations of a group g and. Schur algebras and representation theory book, 1993. Schur algebras are introduced very naturally as algebras of distributions on general linear groups. Calgebras and mackeys theory of group representations. In 8 and 9, henke observed various repeating patterns in the decomposition matrices for the schur algebras s 2, r, and proved the existence of algebra embeddings s 2, r s 2, r for.
The block theory is approached by first proving an analogue of the jantzen sum formula for the \q\ schur algebras. Pdf hecke algebras and schur algebras of the symmetric. Jan 18, 2009 the schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups both finite and infinite. He discusses the usual representation theoretic topics such as. A particular idea leads me to a nearby star on which i decide to land. The q schur algebras were introduced by dipper and james, in the study the representations of hecke algebras and finite general linear groups.
Representation theory of symmetric groups and related. A duality between an affine qschur algebra and a corresponding affine hecke algebra is established. This representation theory was initiated by brauer and it is more algebraic. On the representation dimension of schur algebras miemietz, vanessa. The purpose of this section is to study the relationship of schur algebras and. In mau14, the second author gives a geometric interpretation of the representation theory of the schur algebra skpn,nq in terms of equivariant. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal schur algebras in all characteristics. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra. This leads immediately to the loss of complete reducibility. Martin gives a selfcontained account of this algebra and those links, covering the basic ideas and their quantum analogues. In turn, the representation theory of these latter algebras relates closely to that of the. The qschur algebras were introduced by dipper and james, in the study the representations of hecke algebras and finite general linear groups. In the text dr martin gives a full, selfcontained account of this algebra and these links, covering both the.
A separate part of the book is devoted to each of these areas and they are all treated in sufficient depth to enable and hopefully entice the reader to pursue. Schur indicator, the frobenius divisibility theorem, the burnside the orem, the frobenius. There is a natural duality on quasihereditary algebras, called ringel duality, and the schur algebra skpn,nq is its own ringel dual. Finally, we place these results in the general context of moduli spaces of modules of schurtame algebras. Material not covered here include the theory of induced representations.
Representation theory was born in 1896 in the work of the ger. Adopting a panoramic viewpoint, this book offers an introduction to four different flavors of representation theory. This paper provides a survey for the latest developments in the theory of a. Schur algebras of brauer algebras are defined as endomorphism algebras of certain direct sums of permutation modules over brauer algebras. Combinatorics and representation theory of diagram algebras. The schur algebras and the q schur algebra are finitedimensional algebras which arise in the representation theory of the general linear groups and symmetric groups. Applications of frobenius algebras to representation theory of schur algebras l. Section 5 calculates some examples of schur algebras. We develop a graded version of the theory of cyclotomic qschur algebras, in the spirit of the work of brundankleshchev on hecke algebras and of ariki on qschur algebras. The q schur algebras also have a connection with representations of the finite general. Stuart martin, schur algebras and representation theory, cambridge university press 1993.
We show that the category o for a rational cherednik algebra of type a is equivalent to modules over a qschur algebra parameter. Introduction to representation theory mit opencourseware. Apr 01, 2020 we develop algebraic and geometrical approaches toward canonical bases for affine q schur algebras of arbitrary type introduced in this paper. Very roughly speaking, representation theory studies symmetry in linear spaces. Workshop on representation theory of symmetric groups and related algebras 14 20 december 2017 venue. A sentimental journey through representation theory. Applications of frobenius algebras to representation theory. Explicit combinatorial bases of these new schur algebras are given. Using the machinery of cellular algebras, which is developed in chapter 2, this results in a clean and elegant classification of the irreducible representations of both algebras. Schur algebras and representation theory by martin, stuart. We give a complete classification of the classical schur algebras and the infinitesimal schur algebras which have tame representation type. Schur algebras and representation theory cambridge tracts in. These notes describe the basic ideas of the theory of representations of nite groups.
In the text dr martin gives a full, selfcontained account of this algebra and these links, covering both the basic theory of schur algebras and related areas. Schurs lemma for algebraically closed fields let v be a finite dimensional irreducible representation of an algebra a over an algebraically. The results in the paper were reported at the international asialink conference on algebras and representation theory in beijing normal university in 2005. Tensor products and duals of representations of lie algebras section 2. Quantum schur algebras and their affine and super counterparts. In mathematics, schurs lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In particular it is shown that for fixed n the representation dimensions of the schur algebras get arbitrarily large. We introduce an inner product on the affine q schur algebra, with respect to which the canonical basis is shown to be positive and almost orthonormal. Schur algebras, as endomorphism algebras, connects the representation theory of general linear groups and the representations of symmetric groups via schurweyl duality. Finally, we give an lltlike algorithm for computing the graded decomposition numbers of the quiver schur algebras in characteristic zero when e 0. Another unique feature is an exposition of the relationship between these correspondences, the representation theory of symmetric groups and alternating groups and the theory of symmetric functions.
These results follow from the unicity of certain highest weight categories covering hecke algebras. The third problem concerns the properties of certain generalizations of khovanovlaudarouquier algebras, called quiver schur algebras, and their relationship to cohomological hall. Its general content can be very brie y summarized as follows. The schur algebras and the qschur algebra are finitedimensional algebras which arise in the representation theory of the general linear groups and symmetric groups. Holm, algebras and representation theory, springer. We introduce a generalized version of a qschur algebra of parabolic type for arbitrary hecke algebras over extended weyl groups. We introduce an inner product on the affine qschur algebra, with respect to which the canonical basis is shown to be positive and almost orthonormal. A duality between an affine q schur algebra and a corresponding affine hecke algebra is established. Using a geometric setting for q schur algebras, belinson, lusztig and macpherson realized the quantum enveloping algebra u n of g l n as a limit of q schur algebras over q v. We introduce a generalized version of a q schur algebra of parabolic type for arbitrary hecke algebras over extended weyl groups.
An associative algebra over a eld kis a vector space aover k equipped with an associative bilinear multiplication a. Schur algebras and representation theory, by stuart martin. Goodreads helps you keep track of books you want to read. A schur algebra is a subalgebra of the group algebrargassociated to a partition ofg, wheregis a finite group andris a commutative ring. Schur algebras and representation theory ebook, 1994.
The block theory is approached by first proving an analogue of the jantzen sum formula for the \q\schur algebras. Lecture notes introduction to representation theory. Upon my arrival, i realize that somebody already lives there. Representation type of schur algebras springerlink. As an application, we identify the coe cients of the canonical basis on a higher level fock space with qanalogues of the decomposition numbers of cyclotomic qschur algebras. He discusses the usual representationtheoretic topics such as. Schur algebras are an algebraic system that provide a link between the representation theory of the symmetric and general linear groups.
Schur algebras and representation theory by stuart martin. In this paper, we build up to one of the remarkable results in representation theory called schur. In particular, we give a complete list of the relations that hold in the algebra of schur operators. We apply it to obtain schurweyl duality for representations of gln, c on tensor. We describe how the decomposition matrix of a finite group with split bnpair, with respect to a nondescribing prime. On the representation dimension of schur algebras, algebras. The schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups. This book is the first of its kind covering the topic. The approach is combinatorial where possible, making the presentation accessible to graduate students. Representation type, boxes, and schur algebras julian kulshammer university of stuttgart, germany 10. Mr 2482481, isbn 9780521100465 andrew mathas, iwahorihecke algebras and schur algebras of the symmetric group, university lecture series, vol. Semisimple algebras and double centralizer theorem 19 acknowledgments 20 references 21 introduction. The twoboundary twopole braid group bk is generated by. Using a geometric setting for qschur algebras, belinson, lusztig and macpherson realized the quantum enveloping algebra u n of g l n as a limit of qschur algebras over q v.
We study a representation of the local plactic monoid given by schur operators u i, which act on partitions by adding a box in column i if possible. Explored representation theory of tl k using diagrams and established a connection to the ane hecke algebras of type a and c. As an endomorphism algebra, the complex schur algebra. The third problem concerns the properties of certain generalizations of khovanovlaudarouquier algebras, called quiver schur algebras, and their relationship to cohomological hall algebras. The schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups both finite and infinite.
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