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**Tensor product multiplicities, canonical and totally positive varieties.**
*(English)*
Zbl 1061.17006

The paper under review concludes a project started by the authors in [A. D. Berenshtejn and A. V. Zelevinskij, J. Geom. Phys. 5, No. 3, 453–472 (1988; Zbl 0712.17006)]. The original goal of the project was to find explicit combinatorial expressions for the multiplicities in the tensor product of two simple finite dimensional modules over a complex semisimple Lie algebra in terms of polyhedra, i.e. as the number of lattice points in some convex polytope or, equivalently, as the number of integer solutions of a system of linear equations and inequalities. The conjectured polyhedral expressions were given in the above cited paper of the authors.

In the present paper the authors explicitly construct a family of polyhedral expressions for the tensor product multiplicities. They associate two such expressions to every reduced word of the longest element of the Weyl group. They also produce two universal expressions, the tropical Plücker models.

From the beginning, another important aim of the project was to develop the combinatorial understanding of good bases in finite dimensional representations of semisimple Lie algebras. More precisely, the goal was to understand the combinatorial structure of the Lusztig canonical bases and the Kashiwara global bases in quantum groups. Although the approaches of Lusztig and Kashiwara are equivalent, they lead to different parameterizations. In the paper the authors give an explicit description of the relationship between these parameterizations.

The approach of the authors is based on the remarkable fact observed by Lusztig that combinatorics of canonical bases is closely related to geometry of totally positive varieties. The authors formulate the relationship in terms of two mutually inverse transformations: tropicalization and geometric lifting. The starting point is the observation that different parameterizations are related to each other by piecewise-linear transformations that involve only the operations of addition, subtraction and taking the minimum of two integers.

The paper under review contains important results and gives striking relations between quite different mathematical objects. Although doing with many branches of mathematics, it is well written and may be followed also by nonexperts.

In the present paper the authors explicitly construct a family of polyhedral expressions for the tensor product multiplicities. They associate two such expressions to every reduced word of the longest element of the Weyl group. They also produce two universal expressions, the tropical Plücker models.

From the beginning, another important aim of the project was to develop the combinatorial understanding of good bases in finite dimensional representations of semisimple Lie algebras. More precisely, the goal was to understand the combinatorial structure of the Lusztig canonical bases and the Kashiwara global bases in quantum groups. Although the approaches of Lusztig and Kashiwara are equivalent, they lead to different parameterizations. In the paper the authors give an explicit description of the relationship between these parameterizations.

The approach of the authors is based on the remarkable fact observed by Lusztig that combinatorics of canonical bases is closely related to geometry of totally positive varieties. The authors formulate the relationship in terms of two mutually inverse transformations: tropicalization and geometric lifting. The starting point is the observation that different parameterizations are related to each other by piecewise-linear transformations that involve only the operations of addition, subtraction and taking the minimum of two integers.

The paper under review contains important results and gives striking relations between quite different mathematical objects. Although doing with many branches of mathematics, it is well written and may be followed also by nonexperts.

Reviewer: Vesselin Drensky (Sofia)

### MSC:

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

17B20 | Simple, semisimple, reductive (super)algebras |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

20G05 | Representation theory for linear algebraic groups |

12K10 | Semifields |