Crystal Bases


Spring 2020

  • This is the website for a reading course on the book Crystal Bases: Representations and Combinatorics by Daniel Bump and Anne Schilling. We are Hunter Davenport, Lucas Gagnon, Sarah Salmon, Justin Willson and Tianyuan Xu. We meet 9:30--10:50 on Tuesdays and 11:00--11:50 on Fridays, and we take turns leading the discussions in the meetings.

  • Weekly summaries of our meetings and some informal, incomplete notes (mostly computations of examples) can be found below.

    Week Summary
    1 We started Chapter 2 (Kashiwara crystals, Chapter 1 is the introduction), talked about root systems and weights, and filled in some Lie-theoretic background the book opted to leave out, including computations in type A showing how the root systems and weights arise from the Lie algebras and their natural modules. Another topic not available in the book that we discussed was the restrictions the root system axioms put on the angles between roots and consequences of the restrictions. Notes (Tianyuan) (a slightly different version).
    2 We covered the first half of Chapter 2, culminating in a walk-through of the proof that tensors of crystals are still crystals. Notes (Sarah).
    3 The Tuesday meeting was cancelled due to heavy snow. On Friday we went through a computation on the natural module of the Lie algebra of type B to motivate the crystal axioms. Notes (Tianyuan).
    4 We finished Chapter 2 and did several examples to gain intuition about the tensor rule for crystals and to understand the signature rule. The examples included a two-fold and a three-fold crystal tensor in type A, as well as the row tableaux and column tableaux examples from the book. Notes (Sarah).
    5 We covered Chapter 3 (crystals of tableaux), with an emphasis on computing Kashiwara operators on some example of crystals of tableaux via the row reading embedding of tableaux into tensors powers of the GL(n) crystals. Notes (Hunter).
    6 We started Chapter 4 (Stembridge crystals). Much time was spent on Example 4.1, a verification that the pinched crystal from Figure 3.1 is not Stembridge; during the verification we found a typo in the statement of Lemma 4.4---S2 needs to be included as a hypothesis. We also went through other main results of the chapters, but without going through the proofs carefully. Notes (Lucas).
    7 We tried to deduce the Stembridge axioms from Serre relations in Lie algebras with simply-laced root systems, but did not succeed before we moved on (we might come back to this later). Then we started Chapter 6 (crystals of tableaux II) and finished Section 6.1, which deals with the column reading embedding of type-A tableaux crystals into tensor powers of GL(n) crystals. Notes (Tianyuan).
    8 We started Chapter 7 (insertion algorithms) and talked about Schur-Weyl duality and the RSK algorithm. We didn't meet on Friday due to the COVID-19 outbreak. Notes (Hunter).
    9 We resumed meetings after a two-week break due to COVID-19 and the spring break. On Tuesday we studied the Edelman--Greene algorithm. On Friday we started on Chapter 8 and introduced plactic and Knuth equivalence. Notes (Justin, Sarah).
    10 We studied Chapter 8 and talked about how Knuth equivalence implies plactic equivalence, how RSK relates to these two equivalences, and the connections between crystals of skew tableaux, Littlewood--Richardson coefficients and Levi branching. Our emphasis was doing examples that illustrate the main results. Notes (Lucas).
    11 We discussed exercises 8.3, 8.4 and 8.5, then moved back to Chapter 5 (which we skipped). We covers sections 5.1 and 5.2, on the construction of virtual crystals via embeddings of non-simply-laced roots systems into simply-laced root systems. Notes (Justin).
    12 Professor Schilling, an author of the book we use, gave a Zoom talk at our department this week on applications of crystals to stable Grothendieck polynomials. As preparation we studied Chapter 10, crystals for Stanley symmetric functions, which the talk's topic may be viewed as a variation of. More specifically, we learned how to prove the Schur positivity and interpret the Schur multiplicities of Stanley symmetric functions using crystals, as well as how the morphisms from crystals of decreasing factorizations to crystals of tableaux work via the Edelman--Greene algorithm. All these are generalized in the talk, and the Schur positivity proof gave us a first glimpse at an actual application of crystals. Notes (Tianyuan, Hunter)
    13 This was the last week of our meetings, and we tried to use it to start Chapter 12, which deals with the B_\infty crystal. We learned the construction of B_\infty via elementary crystals in simply-laced types and worked out some examples. We also worked on Exercise 12.1 to see how we may realize B_\infty in type A via large tableaux. We noted that we didn't have time to venture into non-simply-laced theory or the representation theoretic origins of crystals too much in the term, but even so we did learn a lot together, especially about Young tableaux. We hope to look into the non-simply-laced theory and representation theory in some other place, perhaps at some point in this course on Lie algebras (website to be created) . Notes (Lucas, Hunter)