We cite the publisher’s text and from the authors’ introduction:
“This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.”
From the introduction: “In this book, we will limit ourselves to crystals associated to finite-dimensional Lie algebras, omitting the important topic of crystals of representations of infinite-dimensional Lie algebras. Within this limited scope, we have tried to prove the essential facts using combinatorial methods. The facts one wants to prove are as follows.
Given a reductive complex Lie group \(G\), there is an associated weight lattice \(\Lambda\) with a cone of dominant weights. Given a dominant weight \(\lambda\), there is a unique irreducible representation of highest weight \(\lambda\). There are two operations on these that we are particularly concerned with: tensor product of representations and branching, or restriction, to Levi subgroups.
In the theory of crystal bases, one starts with the same weight lattice and cone of dominant weights. Instead of a representation, one would like to associate a special crystal to each dominant weight. If the re presentation is irreducible, the crystal should be connected. There may be many connected crystals with a given highest weight, but it turns out that there is one particular one that we call normal. We think of this as the “crystal of the representation”. More generally, a crystal that is the disjoint union of such crystals, is to be considered normal.
The operations of tensor product and Levi branching from representation theory also make sense for crystals. The usefulness of the class of normal crystals is that the decomposition of a crystal into irreducibles with respect to these operations is again normal. Moreover, the decomposition of a representation obtained by tensoring representations or branching a representation to a Levi subgroup gives the same multiplicities as the decomposition of the tensor product or Levi branching of the corresponding normal crystals into irreducibles.
There are several ways of defining normal crystals. Kashiwara and Littelmann gave two different definitions, which then were shown to be equivalent. We give yet another definition of normal crystals, based on two key ideas: Stembridge crystals and virtual crystals (Kashiwara; Baker). For the simply-laced Cartan types, Stembridge showed how to characterize the normal crystals axiomatically. This is subject of Chapter 4. This approach does not work as well for the non-simply-laced types, but for these, there is a way of embedding certain crystals into crystals of corresponding simply-laced types. For example, to construct a normal \(\mathrm{Sp}(2r)\) crystal (for the non-simply-laced Cartan type \(C_r)\) first one constructs a \(\mathrm{GL}(2r)\) crystal (for the simply-laced Cartan type \(A_{2r-1})\). Then one finds the symplectic crystal as a “virtual crystal” inside the \(\mathrm{GL}(2r)\) one. This way, one may reduce many problems about crystals to the simply-laced case, including the construction of the normal crystals (see Chapter 5). …
A direct approach to the refined Demazure character formula seems difficult, even in the simply-laced case armed with the Stembridge axioms. Instead, what works is to construct the Demazure crystals inside the crystal \(\mathcal B_\infty\). This is an infinite crystal that contains a copy of \(\mathcal B_\lambda\) for every dominant weight \(\lambda\). We construct the Demazure crystals in \(\mathcal B_\infty\) in Chapter 12 and then deduce the properties of their counterparts in \(\mathcal B_\lambda\). After this, we are able to finish the proof of the Demazure character formula for crystals, and thereby establish the relationship with representations as shown in Chapter 13.
The infinite crystal \(\mathcal B_\infty\) is itself a remarkable combinatorial object. It is, in a sense, the crystal of a representation, albeit an infinite-dimensional one, the Verma module with weight 0. As we discover in the proof of the refined Demazure character formula, a clear understanding of \(\mathcal B_\infty\) may be the key to the finite normal crystals. Therefore after we prove the Demazure character formula we investigate \(\mathcal B_\infty\) in more depth. Chapter 14 considers the \(*\)-involution of \(\mathcal B_\infty\), a self-map of order two with remarkable properties that was studied by Lusztig and Kashiwara.
Then in Chapter 15, we turn to another combinatorial realization of B\(\mathcal B_\infty\) that arose from Lusztig’s canonical bases in the theory of quantum groups. In considering the Lusztig realization of \(\mathcal B_\infty\), we begin to see an important theme in this subject: how combinatorial maps that arise in crystal base theory are tropicalizations of algebraic maps. The algebraic maps that admit tropicalizations are those constructed using addition, multiplication and division, but never subtraction. (These are the operations that preserve the positive real numbers, explaining why this topic touches on the theory of total positivity.) Tropicalization replaces the algebraic operations of addition, multiplication and division by piecewise linear ones, and an algebraic map corresponds to a combinatorial one given by piecewise-linear maps. Conversely, given a piecewise-linear map, we may seek an algebraic lifting, an algebraic map having the given piecewise-linear one as its tropicalization. If this is done carefully, the algebraic liftings of various maps may fit together in a way that mirrors the crystal structure
Other topics that we discuss include the crystals of tableaux of Kashiwara and Nakashima, which give explicit models for the normal crystals in the classical Cartan types A, B, C, and D (see Chapters 3 and 6). We discuss the action of the Weyl group on the crystal and the Schützenberger-Lusztig involution.
In Chapters 7 through 10, we specialize to type A to explore various aspects of tableaux theory using crystals. In particular, we discuss Lascoux and Schützenberger’s theory of the plactic monoid and prove the Littlewood-Richardson rule. In the context of this discussion we emphasize analogies between the theory of crystals and representations.
Two appendices on topics in representation theory of \(\mathrm{GL}(n,\mathbb C)\) have been provided to help the reader see these analogies. Roughly, Appendix A on Schur-Weyl duality is analogous to Chapter 8 on the plactic monoid, and the material in Appendix B on the \(\mathrm{GL}(n) \times \mathrm{GL}(m)\) duality is analogous to Chapter 9 on bicrystals and the Littlewood-Richardson rule. On the crystal side, the Robinson-Schensted-Knuth (RSK) insertion algorithm in its various forms plays the role of the Schur-Weyl and \(\mathrm{GL}(n) \times \mathrm{GL}(m)\) dualities.
We have already mentioned that an important role is played in several chapters by reduced words representing the long Weyl group element. In Chapter 15 we will see that for each such reduced word \(\mathbf i\) there is a map \(v\mapsto v_{\mathbf i}\) of the crystal into \(\mathbb N^N\). We will exhibit polytopes called MV polytopes that encode the components of the vector \(v_{\mathbf i}\) in the lengths of various paths around the boundary of the polytope.
Thus reduced words for \(w_0\) are important in studying crystals. In the other direction, at least for type A, Morse and Schilling showed that crystals can be used to study the reduced words for \(w_0\) (or more generally any Weyl group element). As we have already mentioned the Schur function associated to the dominant weight or partition \(\lambda\) can be viewed as the character of the highest weight crystal of highest weight \(\lambda\) in type A. Another important class of symmetric functions are the Stanley symmetric functions (Stanley(1984)), which were introduced to study reduced expressions of symmetric group elements. Stanley symmetric functions have a positive integer expansion in terms of Schur functions. We demonstrate that these can be understood in terms of crystals by imposing a crystal structure on the combinatorial objects underlying the Stanley symmetric functions. In this case, the insertion algorithm of Edelman and Greene (1987), which is a variant of RSK, plays a crucial role. This is done in Chapter 10.”
This book deserves a wide readership!