Buildings and combinatorial representation theory

The American Institute of Mathematics

The following compilation of participant contributions is only intended as a lead-in to the ARCC workshop “Buildings and combinatorial representation theory.” This material is not

for public distribution.

Corrections and new material are welcomed and can be sent to workshops@aimath.org Version: Thu Mar 15 16:51:21 2007

Table of Contents

A. Participant Contributions . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Ballantine, Cristina

2. Goertz, Ulrich

3. Kamnitzer, Joel

4. Karaali, Gizem

5. Kirillov, Alexandre

6. Kumar, Shrawan

7. Kuperberg, Greg

8. Lenart, Cristian

9. McAllister, Tyrrell

10. Millson, John

11. Misra, Kailash

12. Ram, Arun

13. Schilling, Anne

14. Schwer, Christoph

15. Setyadi, Alison

16. Sturmfels, Bernd, and Yu, Josephine

Chapter A: Participant Contributions

A.1 Ballantine, Cristina

I am interested in both directions of the relationship between combinatorics and the representation theory of p-adic groups. Going from representation theory to combinatorics, I am interested in constructing infinite families of Ramanujan graphs and bi-graphs from the Bruhat-Tits building of the respective p-adic group. In the opposite direction, I have recently started to use the combinatorial properties of the building viewed as a graph to construct new p-adic representations of GL_{n}(ℚ_{p}). In both settings, the connection between graph theory and representation theory is made via the Iwahori-Hecke algebra.

In a different context, I am also interested in the tensor product of Schur functions and combinatorial interpretations of Kronecker coefficients.

A.2 Goertz, Ulrich

The motivation for much of my recent work comes from the study of the reduction of Shimura varieties to positive characteristic. This is an important topic in order to understand arithmetic properties of these Shimura varieties and has applications in the Langlands program, for instance. It turns out, that these questions are related in several ways to the geometry of affine Grassmannians and affine flag varieties, and hence to Bruhat-Tits buildings and their combinatorics.

One particular topic with a strong relation to the workshop are the so-called affine Deligne-Lusztig varieties, certain subvarieties of (partial) affine flag varieties, whose properties are to some extent encoded in the combinatorial structures associated with Bruhat-Tits buildings. Some open questions about these varieties can be translated into questions about “folding galleries” or about intersections (in the flag variety) of orbits of certain subgroups [1], similar to those which occur in the work of Gaussent and Littelmann, and also in connection with MV polytopes.

Another instance of such a relationship is provided by the so-called sheaves of nearby cycles (which encode information about the singularities) on the Shimura variety or equivalently a “local model” which sits inside an affine flag variety. This subject also has a strong combinatorial flavor, since it is connected to the Iwahori-Hecke algebra; see [2]. There is also a connection with Ram’s theory of alcove walks [3].

[1] U. Goertz, T. Haines, R. Kottwitz, D. Reuman, Dimensions of some affine Deligne-Lusztig varieties. Ann. sci. de l’ENS. 4e serie, t. 39 (2006), 467-511

[2] U. Goertz, T. Haines, The Jordan-Hoelder series for nearby cycles on some Shimura varieties and affine flag varieties. To appear in J. Reine Angew. Math., math.AG/0402143

[3] U. Goertz, Alcove walks and nearby cycles on affine flag manifolds. math.RT/0610839

A.3 Kamnitzer, Joel

There are two approaches to understanding MV cycles. The first, due to Anderson, involves studying their moment polytopes, called MV polytopes (these polytopes were later combinatorially described in my thesis). The second approach, due to Gaussent-Littelmann, involves resolving the affine Schubert varieties via galleries in the affine building. This leads to the combinatorics of LS galleries, which are closely related to Littelmann paths. It would be very interesting to unify these approaches. Ideally this would be done at the geometric level, but even a nice combinatorial description of the bijection between Littelmann paths and MV polytopes would be desirable.

I am also very interested in learning from Kapovitch and Millson and sharing ideas with them. Their approach to proving saturation conjectures using the metric geometry of buildings has been extremely successful. I would like to understand how to think about their approach in terms of MV cycles and to see what ideas can flow back and forth. The work of Haines is an important link between these two worlds.

A.4 Karaali, Gizem

As a novice in the area I am not certain exactly which open problems may be of interest to the workshop participants. However I expect to learn a lot and hope to contribute to the discussions throughout the week as I get my bearings.

A.5 Kirillov, Alexandre

This year I give a course “Root systems and Coxeter groups at UPenn and run a seminar Representation theory. I believe that participating in the Workshop will be profitable for both.

My own plans inclute the continuation of the sturdy of so-called Family algebras. They are natural examples of free U(𝔤)-motules of finite rank which have a structure of Z(𝔤)-algebras. This study was initiated in my paper

1. Family algebras Electronic Research Announcements of AMS, vol. 6 No 1, 2000. and continued in

2. Kirillov A.A. Some results on the Strucure of Quantum Family Algebras, Preprint ESI No 964, November 27, 2000.

3. Kirillov A.A. Introduction to Family Algebras, Moscow Math. Jour. vol.1, Num.1, Jan-March 2001, pp. 27-41.

4. Rojkovskaya N.A. PhD Thesis at UPenn, 2001

5. Kirillov A.A., Rybnikov L. Odd family algebras, In: Studies in Lie Theory, Progress in Math., vol. 243 Birkhäuser (2006), pp.303-318

6. Panyushev D.I. Weight multiplicity free representations, ^{\underline{g}}-endomorphism algebras, and Dynkin polynomials, Preprint arXiv:math.AG/0112314v2 2 Apr 2002

A.6 Kumar, Shrawan

My main interests lie in the representation theory of (finite dimensional) semisimple Lie algebras and their quantized versions and also the Kac-Moody algebras. Currently I am interested in the saturation problem for the tensor product of irreducible representations. The recent works of Kapovich-Leeb-Millson have used the geometry of Buildings to give a solution of the problem for general semisimple Lie algebras. This work has prompted me to understand the geometry of Buildings in connection with the set of problems motivated by the saturation problems.

A.7 Kuperberg, Greg

I am interested in connections between possible connections between buildings and polygon varieties on the one hand and planar tensor calculus models for invariant theory of low-rank Lie algebras on the other hand. The rank 1 case is now called “spin networks”, and was developed by Newman-Penrose, Temperley-Lieb, Jones-Wenzl, and Kauffman. The rank 2 case was done by me, and recently Westbury and Kim have made progress on the rank 3 case. One ultimate goal is to have a suitable presentation of the invariant tensors over any simple Lie algebra as a pivotal category, and it seems likely that the geometry of buildings will help with this problem.

Another interesting problem is the categorification program of Khovanov et al. It has already been connected to the geometry of buildings in light of the paper of Cautis and Kamnitzer, for example.

A.8 Lenart, Cristian

Background

In the joint papers with A. Postnikov [la1,la2], we developed a type-independent combinatorial model, called the alcove path model, for the irreducible representations of semisimple Lie algebras and, more generally, of complex symmetrizable Kac-Moody algebras. There are other models in this area, such as: Kashiwara-Nakashima tableaux [ka1], Littelmann paths [li1,li2,li3], LS-galleries [gallgp], the model in [ba1] based on Lusztig’s parametrization of canonical bases, the Kyoto path model [kk1,kk2], some models based on geometric constructions etc. Ours has advantages related to its generality, simplicity, combinatorial nature, and other applications, such as Demazure modules (which form a filtration of the irreducible modules) and Schubert calculus. In particular, it leads to a far-reaching generalization of the type A combinatorics of Young tableaux [fu1,ll1,st22]. I have also developed a Maple software package for manipulations based on the alcove path model [le1].

I will now briefly explain the constructions involved in the alcove path model. Let 𝔤 be a complex simple Lie algebra with Cartan subalgebra 𝔥. Let Φ ⊂ 𝔥^{∗} be its root system with positive roots Φ^{+}, and let 𝔥_{ℝ}^{∗} ⊂ 𝔥^{∗} be the real span of Φ. The nondegenerate scalar product on 𝔥_{ℝ}^{∗} induced by the Killing form is denoted by ⟨ · , · ⟩. Given α ∈ Φ (with coroot α^{∨}) and an integer k, let H_{α,k} be the hyperplane {λ ∈ 𝔥_{ℝ}^{∗} : ⟨λ, α^{∨}⟩ = k}. The connected components of 𝔥_{ℝ}^{∗} \ (⋃_{α,k} H_{α,k}) are called alcoves. The fundamental alcove is A_{∘} := {λ ∈ 𝔥_{ℝ}^{∗} : 0 < ⟨λ, α^{∨}⟩ < 1, α ∈ Φ^{+}}. For a pair of adjacent alcoves (i.e., distinct and having a common wall), we write A ⟶^{β} B if the common wall is of the form H_{β,k} and the root β points from A to B. Given a weight λ, let A_{λ} be the translation of A_{∘} by λ. The Bruhat order on the Weyl group W is defined by its covers w ⋖ ws_{α}, for α ∈ Φ^{+}, if ℓ(ws_{α}) = ℓ(w) + 1 (as usual, s_{α} is a reflection and ℓ(w) is the length of w).

Definition 1. An alcove path is a sequence of alcoves (A_{0}, A_{1}, …, A_{n}) such that A_{i−1} and A_{i} are adjacent, for i = 1, …, n. We say that an alcove path is reduced if it has minimal length among all alcove paths from A_{0} to A_{n}. Given a dominant weight λ and a reduced alcove path A_{∘} = A_{0} ⟶^{−β_{1}} ⋯ ⟶^{−β_{n}} A_{n} = A_{−λ}, we call the sequence of (positive) roots (β_{1}, …, β_{n}) a λ-chain.

Let us fix a λ-chain Γ = (β_{1}, …, β_{n}), and let r_{i} be the reflection s_{β_{i}}. Consider the alcove path (A_{0}, A_{1}, …, A_{n}) corresponding to Γ, and let r̂_{i} be the affine reflection in the hyperplane containing the common wall of the alcoves A_{i−1} and A_{i} in the alcove path. We now define the centerpiece of the alcove path model, which is our generalization of semistandard Young tableaux in type A.

Definition 2. An admissible subset is a subset of [n] := {1, …, n}, that is, J = {j_{1} < j_{2} < … < j_{s}}, such that we have the following saturated chain in the Bruhat order on the Weyl group W:

1 ⋖ r_{j_{1}} ⋖ r_{j_{1}}r_{j_{2}} ⋖ … ⋖ r_{j_{1}}r_{j_{2}} … r_{j_{s}} .

Let A(Γ) be the collection of admissible subsets corresponding to the λ-chain Γ. Given J ∈ A(Γ), we define its weight μ(J) and key κ(J) (by analogy with the Lascoux-Schützenberger key [la3]) by

μ(J) := −r̂_{j_{1}} … r̂_{j_{s}}(−λ) , κ(J) := r_{j_{1}} … r_{j_{s}} ∈ W .  (1)

Given a dominant weight λ and an element w of the Weyl group W, we have the Demazure module V_{w}(λ). It is well-known that the Demazure module V_{w_{∘}}(λ) for the longest element w_{∘} of W coincides with the irreducible module V (λ) of 𝔤 with highest weight λ. The character formula below shows that the set A(Γ) is an indexing set for a basis of an irreducible representation.

Theorem 1 (le2,la1). We have the following Demazure character formula:

ch(V_{w}(λ)) = ∑_{\begin{matrix}J∈A(Γ) \\ κ(J)≤w \end{matrix}} e^{μ(J)} .

Kashiwara [ka2,ka3] defined a directed colored graph, called crystal graph, on the canonical basis of an irreducible representation [ka2,lu1]. This graph partially encodes the action of the Chevalley generators of 𝔤 on the canonical basis.

In [la2], we defined combinatorially partial operators E_{i}, F_{i} on A(Γ) associated to the simple roots α_{i} and called root operators. They define a directed colored graph (and a poset) on A(Γ) with arrows (covers) x → y colored i for F_{i}(x) = y. The connection to crystal graphs is as follows.

Theorem 2 (le2,la2). For any λ-chain Γ, the graph on A(Γ) defined by the root operators is isomorphic (as a directed colored graph) to the crystal graph of the irreducible representation V (λ).

Remark. I sometimes use the notation A(λ) for A(Γ), assuming that there is a choice of a λ-chain Γ. According to Theorem 2, it makes sense to call the set A(λ) a crystal.

The representation-theoretic applications of the alcove path model that I developed so far, partly in joint work with A. Postnikov (in [le2,la2]), are mainly related to the combinatorics of crystal graphs, and are mentioned below.

• The generalization of the model to complex symmetrizable Kac-Moody algebras.

• A Littlewood-Richardson rule for decomposing tensor products of irreducible modules.

• A type-independent combinatorial realization of Lusztig’s involution [lu2] on the canonical basis of an irreducible representation, exhibiting A(Γ) as a self-dual poset; this is the first direct generalization of Schützenberger’s involution on Young tableaux [fu1];

• The definition of certain type-independent combinatorial moves which generalize Schützenberger’s sliding algorithm, also known as jeu de taquin [fu1]; these moves were called Yang-Baxter moves (being based on the Yang-Baxter equation), and have several applications.

Interests

I am interested in studying the relationship between the alcove path model and some of the other models for crystals (such as tableaux, the Kyoto path model etc.), thus providing a dictionary between them. I am also interested in potential applications of the alcove path model or of other models to:

• a type-independent approach to the theory of the plactic algebra, as developed by Lecouvey in the classical types [le3,le4,le5], cf. also [li4,ll1] (the plactic algebra is a noncommutative model for the representation ring);

• a combinatorial formula for the Kostka-Foulkes polynomials K_{λμ}(q), which are qanalogs of weight multiplicities in the irreducible representations [lu3,ma1] (in type A, they are described by the charge statistic on Young tableaux [la4], while Lecouvey used a statistic on Kashiwara-Nakashima tableaux [ka1] to compute certain K_{λμ}(q) in types B–D [le7]);

• explicit basis constructions of the irreducible representations (or of the Demazure modules) for the semisimple Lie algebras (or the symmetrizable Kac-Moody algebras);

• using the commutor^{1} for crystals defined by Henriques and Kamnitzer [ha1] (see also [ka4,le22] and the section below), or perhaps a different method, in order to give a reasonable definition/construction for the “symmetric power of a crystal”; it is expected that such a construction corresponds to some q-deformation of the ordinary symmetric power of a module, such as that in [ba2,ba3].

Proposed problem

It is well-known that the category 𝔤-Crystals of crystals for complex semisimple Lie algebras is a monoidal category with an associative tensor product (e.g., see [ha1]). The crystals X ⊗ Y and Y ⊗ X are isomorphic (via maps called commutors), but (x, y) ↦ (y, x) is not a commutor. Henriques and Kamnitzer [ha1] defined a commutor σ_{X,Y} based on an idea of Berenstein. More precisely, if η_{X} is Lusztig’s involution on the crystal X, then

σ_{X,Y} (x, y) := η_{Y ⊗X}(η_{Y} (y), η_{X}(x)) .  (2)

Theorem 3 (ha1). 𝔤-Crystals with the above commutor is a coboundary category (cf. [dr1]).

Kamnitzer and Tingley [ka4] proved that the action of this commutor on the highest weight elements (which determines it) is given by Kashiwara’s involution on the Verma crystal.

Let us turn to the alcove path model. Specifying a commutor for the crystals X := A(λ) and Y := A(μ) amounts to constructing a weight-preserving bijection between the sets max X ⊗ Y and max Y ⊗ X of maximal elements in the two product crystals (the weight is defined by μ(J, K) = μ(J) + μ(K), cf. also (1)). Since the elements of max X ⊗ Y are of the form (∅, K), specifying a commutor amounts to constructing an appropriate partial map K ↦ J from Y to X. The following striking feature of the commutor in [ha1], which is intrinsic to the alcove path model and is related to Demazure characters (cf. Theorem 1), was thoroughly checked experimentally:

κ(J) = κ(K)^{−1} in the Weyl group W, for K ↦ J .  (3)

1Henriques and Kamnitzer use the term commutator. The above terminology was suggested to me by A.

Knutson.

Problem. Check whether (3) is true in general, and, if so, find an explicit combinatorial realization of σ_{X,Y} that explains it.

Remarks. (1) The case corresponding to λ and μ being minuscule or quasi-minuscule weights case is solved by some moves defined by van Leeuwen [le6], cf. the remark below.

(2) I gave an explicit combinatorial realization of σ_{X,Y} in [le22]. Instead of being based on the sets A(λ) and A(μ) themselves, it is based on their embedding into a tensor product of crystals corresponding to minuscule or quasi-minuscule weights [?, Theorems 7.1, 7.6], as well as on the combinatorial moves in [le6]. However, this approach offers no indication related to (3). In addition to this aspect, the advantage of the approach proposed above is that, unlike the one in [le22], it is well suited for an extension to the Kac-Moody case; this extension was stated as a problem in [ha1], given that the construction of σ_{X,Y} in [ha1] does not work in the infinite case (note that the construction in [ka4] works in the infinite case).

Bibliography

[ba1] A. Berenstein and A. Zelevinsky. Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math., 143:77–128, 2001.

[ba2] A. Berenstein and S. Zwicknagl. Braided symmetric and exterior algebras. arXiv:math.QA/0504155.

[ba3] S. Zwicknagl. Braided symmetric and exterior algebras II: Flat modules and Poisson structures. In preparation.

[dr1] V. G. Drinfel’d. Quasi-Hopf algebras. Leningrad Math. J., 1:1419–1457, 1990.

[fu1] W. Fulton. Young Tableaux, volume 35 of London Math. Soc. Student Texts. Cambridge Univ. Press, Cambridge and New York, 1997.

[gallgp] S. Gaussent and P. Littelmann. LS-galleries, the path model and MV-cycles. Duke Math. J., 127:35–88, 2005.

[ha1] A. Henriques and J. Kamnitzer. Crystals and coboundary categories. Duke Math. J., 132:191–216, 2006.

[ka4] J. Kamnitzer and P. Tingley. A definition of the crystal commutator using Kashiwara’s

involution.arXiv:math.QA/0610952.

[kk1] S.-J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki. Affine crystals and vertex models. In Infinite analysis, Part A, B (Kyoto, 1991), volume 16 of Adv. Ser. Math. Phys., pages 449–484. World Sci. Publishing, River Edge, NJ, 1992.

[kk2] S.-J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki. Perfect crystals of quantum affine Lie algebras. Duke Math. J., 68:499–607, 1992.

[ka2] M. Kashiwara. Crystalizing the q-analogue of universal enveloping algebras. Commun. Math. Phys., 133:249–260, 1990.

[ka3] M. Kashiwara. On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J., 63:465–516, 1991.

[ka1] M. Kashiwara and T. Nakashima. Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra, 165:295–345, 1994.

[la4] A. Lascoux and M.-P. Schützenberger. Sur une conjecture de H. O. Foulkes. C. R. Acad. Sci. Paris Sér. I Math., 288:95–98, 1979.

[la3] A. Lascoux and M.-P. Schützenberger. Keys and standard bases. In D. Stanton, editor, Invariant Theory and Tableaux, volume 19 of The IMA Vol. in Math. and Its Appl., pages 125–144, Berlin-Heidelberg-New York, 1990. Springer-Verlag.

[le3] C. Lecouvey. Schensted type correspondence for type G_{2} and computation of the canonical basis of a finite dimensional U_{q}(G_{2})-module. arXiv:math.CO/0211443.

[le4] C. Lecouvey. Schensted-type correspondence, plactic monoid, and jeu de taquin for type C_{n}. J. Algebra, 247:295–331, 2002.

[le5] C. Lecouvey. Schensted-type correspondences and plactic monoids for types B_{n} and D_{n}. J. Algebraic Combin., 18:99–133, 2003.

[le6] M. A. A. van Leeuwen. An analogue of jeu de taquin for Littelmann’s crystal paths. Sém. Lothar. Combin., 41:Art. B41b, 23 pp. (electronic), 1998.

[le7] C. Lecouvey. Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems B_{n}, C_{n} and D_{n}. Eur. J. Comb., 27:526–557, 2006.

[le22] C. Lenart. On the combinatorics of crystal graphs, II. The crystal commutor. arXiv:math.RT/0611444.

[le1] C. Lenart. Alcove path, 2006. Maple package for the alcove path model, math.albany.edu/math/pers/lenart/articles/soft.html.

[le2] C. Lenart. On the combinatorics of crystal graphs, I. Lusztig’s involution. Adv. Math., doi:10.1016/j.aim.2006.08.002, 2006.

[la1] C. Lenart and A. Postnikov. Affine Weyl groups in K-theory and representation theory.

arXiv:math.RT/0309207.

[la2] C. Lenart and A. Postnikov. A combinatorial model for crystals of Kac-Moody algebras. arXiv:math.RT/0502147. To appear in Trans. Amer. Math. Soc.

[li1] P. Littelmann. A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras. Invent. Math., 116:329–346, 1994.

[li2] P. Littelmann. Paths and root operators in representation theory. Ann. of Math. (2), 142:499–525, 1995.

[li4] P. Littelmann. A plactic algebra for semisimple Lie algebras. Adv. Math., 124:312–331, 1996.

[li3] P. Littelmann. Characters of representations and paths in 𝔥_{ℝ}^{∗}. In Representation theory and automorphic forms (Edinburgh, 1996), volume 61 of Proc. Sympos. Pure Math., pages 29–49. Amer. Math. Soc., Providence, RI, 1997.

[ll1] M. Lothaire. Algebraic Combinatorics on Words, chapter The plactic monoid (by A. Lascoux, B. Leclerc, and J-Y. Thibon), pages 144–172. Cambridge University Press, Cambridge, 2002.

[lu3] G. Lusztig. Singularities, character formulas, and a q-analog of weight multiplicities. In Analysis and topology on singular spaces, II, III (Luminy, 1981), volume 101 of Astérisque, pages 208–229. Soc. Math. France, Paris, 1983.

[lu1] G. Lusztig. Canonical bases arising from quantized enveloping algebras. II. Progr. Theoret. Phys. Suppl., 102:175–201, 1991.

[lu2] G. Lusztig. Introduction to Quantum Groups, volume 110 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1993.

[ma1] I. G. Macdonald. Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford University Press, Oxford, second edition, 1995.

[st22] R. P. Stanley. Enumerative Combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.

[st1] J. R. Stembridge. Combinatorial models for Weyl characters. Adv. Math., 168:96–131,

A.9 McAllister, Tyrrell

My interest in this workshop centers around the applications of techniques from polyhedral geometry to representation theory. The work of Berenstein and Zelevinsky (2001), Knutson and Tao (1999), and Littelmann (1994) has shown that a polyhedral approach can yield deep results on the behavior of tensor product multiplicities. These results have been significant both from a theoretical perspective and from the perspective of computational complexity. Here are two questions/conjectures motivated by taking a polyhedral approach to tensor product multiplicities.

Question/Conjecture 1:

Fix dominant weights λ, μ, and ν for one of the classical Lie algebras. Let C_{λμ}^{ν} be the multiplicity with which the irreducible representation V_{ν} appears in the tensor product V_{λ}⊗V_{μ}. Stretching each dominant weight by an integral factor n yields the so-called stretched Clebsch–Gordan coefficient C_{nλ,nμ}^{nν}. It is known that C_{nλ,nμ}^{nν} is a quasi-polynomial function of n. That is, there are periodic functions c_{0}, …, c_{d} : ℤ → ℚ such that

C_{nλ,nμ}^{nν} = c_{d}(n)n^{d} + c_{d−1}(n)n^{d−1} + ⋯ + c_{1}(n)n + c_{0}(n).

We conjecture that the coefficient functions c_{i} are nonnegative. In the type-A case, this would imply the Saturation Theorem of Knutson and Tao because, as shown by Derksen and Weyman (2002), C_{nλ,nμ}^{nν} is in this case simply a polynomial function of n.

De Loera and I (2006) have found abundant experimental evidence for this conjecture using the interpretation due to Berenstein and Zelevinsky (2001) of C_{λμ}^{ν} as the number of integer lattice points in a particular polytope P. The stretched Clebsch–Gordan coefficient above then becomes the Ehrhart quasi-polynomial of P, which counts the number of lattice points in its n-times dilation nP. The lattice point enumeration algorithm of Barvinok (1994), which is polynomial in fixed dimension, therefore allows us to compute such quasipolynomials efficiently in low rank.

Question/Conjecture 2:

As mentioned above, polyhedral algorithms allow us to compute a tensor product multiplicity in polynomial time when the rank is fixed. But suppose that we only want to determine whether V_{ν} appears in V_{λ} ⊗ V_{μ}, without regard to its multiplicity. Due to the Saturation Theorem of Knutson and Tao, it turns out that we can answer this question for type A in polynomial time for arbitrary rank. It remains an open question whether this polynomiality result extends to the other classical root systems. Since the Saturation Theorem establishes this result in type A, a reasonable place to start is by finding the right version of that theorem for the other root systems, which is, of course, one of the explicit goals of this workshop.

A.10 Millson, John

I intend to explain how methods from the metric geometry of symmetric spaces and Euclidean buildings can be applied to attack problems in algebraic group theory. This is joint work with Misha Kapovich and fits into a broader context of an NSF FRG grant shared with Prakash Belkale, Tom Haines, Misha Kapovich and Shrawan Kumar (all four will be participants at the conference). Let G be reductive group which is split over ℚ. We will first study saturation problems for the structure constants of the representation ring of the Langlands’ dual G^{∨} and spherical Hecke algebras associated to G and their connection with the geometry problem of constructing triangles in a symmetric space or Euclidean building with given cone-valued side-lengths (the cone is the positive Weyl chamber). Roughly speaking the solution sets of the three problems are related by Rep ⊂ Hecke ⊂ T riangle. One can reverse the inclusions in each case by multiplying the side-lengths by an explicit “saturation factor”, the LCM of the coefficients of the highest root when expressed in terms of the simple roots (this is one or two for the classical groups).

There is a parallel theory for restricting representations to a Levi subgroup L^{∨} of a standard parabolic subgroup P^{∨} of G^{∨} and the constant term map from the Hecke algebra of G to the Hecke algebra of L. This is joint work with Tom Haines and Misha Kapovich. Now the associated geometry problem in a symmetric space or Euclidean building is the problem of constructing an ideal quadrilateral with one fixed vertex at infinity (a point fixed by P) with two fixed cone-valued side-lengths and two-fixed cone-valued ideal sidelengths (values of the cone-valued Busemann function). Once again one has the chain of implications Rep ⊂ Hecke ⊂ IdealT riangle and one can reverse the arrows by multiplying the side-lengths by the same saturation factors as above.

A.11 Misra, Kailash

I am looking forward to this workshop to learn more about the combinatorial representation theory. In particular, I hope there will be some talks about the geometric crystal bases introduced by A. Berenstein and D. Kazhdan in 2000. Recently, Kashiwara, Okado and others have established an interesting correspondence between the algebraic crystals and geometric crystals for quantum affine algebras via ”ultra-discretization”. I would like to have a better understanding of this correspondence.

A.12 Ram, Arun

My interest in the workshop comes from my work on affine Hecke algebras and representation theory: See my article math.RT/0601343

Some issues I would like the workshop to resolve for me are:

(a) What is the bijection between Littelmann path models and MV polytopes?

(b) What, is the image of a point in the affine flag variety under the moment map?

(c) There are constructions of MV-cycles by GGMS intersections (see Kamnitzer’s paper), by sequences of “root operators” for U^{−}(see Baumann-Gaussent) and by sequences of “root operators” for U^{+} (see Morier-Genoud) and by alcove walks (see Gaussent-Littelmann). In each case one takes a closure to get the MV cycles. What points are added to the original set when one takes the closure? How do the nonclosed sets sit in the loop Grassmanian?

A.13 Schilling, Anne

Crystal bases have been one of the central themes of my research in recent years. I mostly study their combinatorial and representation theoretic properties, and their applications to statistical mechanics, conformal field theory and symmetric functions. I hope to gain a better insight into the geometric aspects of crystals at this workshop.

Some (open) questions, that would be worthwhile to address at the workshop, are:

(1) establish the precise relation between the alcove path model and the tableaux realization of crystals;

(2) find the analogue of charge on tableaux or the energy function on Kyoto paths in the geometric setting;

(3) in a recent paper with Fourier and Shimozono, we gave an algorithm for the computation of the combinatorial R-matrix for affine finite-dimensional crystals. This involves Demazure crystals and dual crystals (corresponding to the Lusztig involution). It would be desirable to find the geometric interpretation of this construction.

(4) is there an analogue of the saturation conjecture/theorem for fusion coefficients?

A.14 Schwer, Christoph

I am interested in the gallery model of Gaussent–Littelmann and its relations to the combinatorial description of q-analogs of symmetric functions for all types like Macdonald polynomials. For this I would be interested in a building-theoretical definition of the double affine Hecke algebra.

A.15 Setyadi, Alison

I have studied the combinatorics of the affine buildings Ξ_{n} and Δ_{n} naturally associated to the special linear group of degree n and to the symplectic group of degree n, respectively, over a local field K with finite residue field. In particular, I used a natural embedding of Δ_{n} in Ξ_{2n} not only to prove results about Δ_{n} using results about Ξ_{2n}, but also to adapt the proofs of certain results about Ξ_{2n} to prove analogous results about Δ_{n}. The role of buildings in combinatorial representation theory is a topic I would like to explore as a direction in which to take my research.

A.16 Sturmfels, Bernd, and Yu, Josephine

Abstract of the paper ”Affine Buildings and Tropical Convexity” by Joswig, Sturmfels, and Yu.

The notion of convexity in tropical geometry is closely related to the notion of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat-Tits building of SL_{d}(K), and techniques for computing with apartments and membranes. While our original inspiration was the work of Dress and Terhalle in phylogenetics, and the work of Keel and Tevelev in algebraic geometry, the tropical approach is applicable also in p-adic geometry and representation theory.