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Zbl 1003.17008
Ariki, Susumu
Representations of quantum algebras and combinatorics of Young tableaux. Transl. from the Japanese and revised from the author. (English)
[B] University Lecture Series. 26. Providence, RI: American Mathematical Society (AMS). vii, 158 p. \$ 31.00 (2002). ISBN 0-8218-3232-8/pbk

Suppose that $b=(\varepsilon_1,\varepsilon_2,\ldots ,\varepsilon_n)$ is a sequence of plus and minus signs. Then there are two well-defined operations that can be carried out on $b$. We cross out all pairs $(+,-)$ occurring in $b$ and repeat this process until there are no such pairs left; call the sequence with signs crossed out $\widetilde{b}$. Let $r$ be the position of the rightmost minus sign in $\widetilde{b}$ and let $s$ be the position of the leftmost plus sign in $\widetilde{b}$ (keeping the numbering of $b$). Define $e(b)$ to be $b$ with the sign of $\varepsilon_r$ changed to plus, and define $f(b)$ to be $b$ with the sign of $\varepsilon_s$ changed to minus. \par This combinatorial device appears in two seemingly different contexts: in Kleshchev's modular branching rule for the symmetric group, in which case the sequence $b$ appears as a sequence of addable and removable nodes of a Young tableau associated with a residue [see {\it A. S. Kleshchev}, J. Algebra 178, 493-511 (1995; Zbl 0854.20013), J. Reine Angew. Math. 459, 163-212 (1995; Zbl 0817.20009), J. Lond. Math. Soc. (2) 54, 25-38 (1996; Zbl 0854.20014) and J. Algebra 201, 547-572 (1998; Zbl 0931.20014)] and in the action of Kashiwara's operators on the crystal graph [{\it M. Kashiwara}, Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)] for the module of an affine Lie algebra of type $A_{r-1}^{(1)}$ of highest weight given by the first fundamental weight $\Lambda_0$. (See also {\it T. Nakashima} and {\it M. Kashiwara} [J. Algebra 165, 295-345 (1994; Zbl 0808.17005)] for explict examples of this combinatorics.) \par This book can be seen as an explanation of this coincidence, which of course is not a coincidence after all, and relies on deep connections between the representation theory of symmetric groups and Hecke algebras on the one side, and the crystal graphs of modules for affine quantized enveloping algebras on the other. \par The book has two main parts. The first part, chapters 1-9, is an introduction to the theory of quantum groups (the quantum algebras of the title). Some background in the theory of simple Lie algebras and their classification by Dynkin diagrams [see e.g. {\it H. Samelson}, Notes on Lie algebras, Springer-Verlag, New York (1990; Zbl 0708.17005)], and Kac-Moody Lie algebras, would be useful to the reader, but a good explanation is given also in these areas in order to understand the correspondence mentioned above. The author gives a good introduction to the algebraic aspects of this fast-developing field, including the theory of the global crystal, or canonical basis, for a quantized enveloping algebra associated to a symmetrizable Kac-Moody Lie algebra as developed by {\it G. Lusztig} [J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008)] and M. Kashiwara (reference as above). The example $A_{r-1}^{(1)}$ (affine type) is used throughout, which works well -- the reader is able to see how things work first hand, as well as obtaining the explicit case needed for the correspondence with modular representation theory. \par In the second part, the author discusses the connection between the modular representation theory of the symmetric group (and the representation theory of Hecke algebras) and the canonical and crystal bases mentioned above. Let $U$ denote the quantized enveloping algebra of type $A_{r-1}^{(1)}$. The author describes the Hayashi realisation [{\it T. Hayashi}, Commun. Math. Phys. 127, 129-144 (1992; Zbl 0701.17008) and {\it K. C. Misra} and {\it T. Miwa}, Commun. Math. Phys. 134, 79-88 (1990; Zbl 0724.17010)] of the module $V(\Lambda_0)$ of $U$ of highest weight $\Lambda_0$ (the first fundamental weight) in terms of Young tableaux, in fact giving a new proof. He shows how this can be used to obtain a realisation of an arbitrary highest weight $U$-module via multipartitions. He goes on to show how this can be used in the $\Lambda_0$ case to obtain a description of the crystal basis of $V(\Lambda_0)$ in terms of the combinatorics of good nodes appearing in the modular representation theory of symmetric groups (a result of K. C. Misra and T. Miwa) as well as discussing the general case. \par Next, the author describes the LLT algorithm [{\it A. Lascoux, B. Leclerc} and {\it J.-Y. Thibon}, Commun. Math. Phys. 181, 205-263 (1996; Zbl 0874.17009)], for obtaining a canonical basis element of $V(\Lambda_0)$ and their conjecture that the coefficients obtained in a decomposition of such an element in terms of the natural tableau basis arising from the Hayashi realisation are decomposition numbers for the Hecke algebra with parameter $q$ given by a primitive $r$th root of unity. The key theorem (12.5 in the book) is the author's generalisation of this conjecture to the Hecke algebras $H_n$ of type $G(m,1,n)$. The remainder of the book is devoted to the author's proof of this result and its important applications. Specifically, he shows that, for an appropriate choice of the $m$ parameters (powers of $q$), the direct sum of the complexified Groethendieck groups of the $H_n$ affords a structure as module for $U$ where the Chevalley generators act as certain restriction and induction operators. Moreover, the module so obtained is a highest weight module (with highest weight obtained from the exponents in the parameters), and if the characteristic of the field of definition of $H_n$ is zero, the canonical basis of this module specialised at $q=1$ coincides with the basis given by the indecomposable projective modules. The proof involves perverse sheaves and the Hall algebra of the cyclic quiver. \par Remark: For an alternative approach to induction and restriction operators on these Grothendieck rings and the link with quantum groups, see also {\it I. Grojnowksi}, preprint arXiv:math.RT/ 9907129). \par Overall, this is a well-written and clear exposition of the theory needed to understand the latest advances in the theory of the canonical/global crystal basis and the links with the representation theory of symmetric groups and Hecke algebras. \par The book finishes with an extensive bibliography of papers, which is well organised into different areas of the theory for easy reference.
[Robert Marsh (Leicester)]

MSC 2000:: *17B37 Quantum groups and related deformations
05E10 Tableaux, etc.
17-02 Research monographs (nonassoc. rings and algebras)
20C08 Hecke algebras and their representations
17B67 Kac-Moody algebras
20C33 Representations of finite groups of Lie type
16D90 Module categories (assoc. rings and algebras)
16G20 Representations of quivers and partially ordered sets
14M15 Grassmannians, Schubert varieties
81R50 Quantum groups and related algebraic methods in quantum theory

Keywords: quantized enveloping algebra; affine Lie algebra; Young tableau; crystal basis; canonical basis; Hayashi realisation; Hecke algebra; LLT algorithm; Kleshchev branching rules; modular representation theory; symmetric group; combinatorial representation theory

Citations: Zbl 0854.20013; Zbl 0817.20009; Zbl 0854.20014; Zbl 0931.20014; Zbl 0739.17005; Zbl 0808.17005; Zbl 0708.17005; Zbl 0703.17008; Zbl 0724.17010; Zbl 0874.17009; Zbl 0701.17008

Cited in: Zbl 1080.20011 Zbl 1067.17506

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