ALGEBRAIC COMBINATORICS

no. 2

On the double-affine Bruhat order: the ε=1 conjecture and classification of covers in ADE type
Algebraic Combinatorics, Volume 2 (2019) no. 2, pp. 197-216.

For any Kac–Moody group G, we prove that the Bruhat order on the semidirect product of the Weyl group and the Tits cone for G is strictly compatible with a ℤ-valued length function. We conjecture in general and prove for G of affine ADE type that the Bruhat order is graded by this length function. We also formulate and discuss conjectures relating the length function to intersections of “double-affine Schubert varieties”.

Received: 2018-01-23
Revised: 2018-05-24
Accepted: 2018-07-23
Published online: 2019-03-05
DOI: https://doi.org/10.5802/alco.37
Classification: 05E10
Keywords: Kac–Moody groups, double-affine Bruhat order 
@article{ALCO_2019__2_2_197_0,
     author = {Muthiah, Dinakar and Orr, Daniel},
     title = {On the double-affine Bruhat order: the $\varepsilon =1$ conjecture and classification of covers in ADE type},
     journal = {Algebraic Combinatorics},
     pages = {197--216},
     publisher = {MathOA foundation},
     volume = {2},
     number = {2},
     year = {2019},
     doi = {10.5802/alco.37},
     mrnumber = {3934828},
     zbl = {1414.05304},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_2_197_0/}
}
Muthiah, Dinakar; Orr, Daniel. On the double-affine Bruhat order: the $\varepsilon =1$ conjecture and classification of covers in ADE type. Algebraic Combinatorics, Volume 2 (2019) no. 2, pp. 197-216. doi : 10.5802/alco.37. https://alco.centre-mersenne.org/item/ALCO_2019__2_2_197_0/

[1] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Volume 231, Springer, 2005, xiv+363 pages | MR 2133266 | Zbl 1110.05001

[2] Braverman, Alexander; Finkelberg, Michael Pursuing the double affine Grassmannian. I. Transversal slices via instantons on A k -singularities, Duke Math. J., Volume 152 (2010) no. 2, pp. 175-206 | Article | MR 2656088 | Zbl 1200.14083

[3] Braverman, Alexander; Kazhdan, David; Patnaik, Manish M. Iwahori-Hecke algebras for p-adic loop groups, Invent. Math., Volume 204 (2016) no. 2, pp. 347-442 | Article | MR 3489701 | Zbl 1345.22011

[4] Finkelberg, Michael; Mirković, Ivan Semi-infinite flags. I. Case of global curve ℙ 1 , Differential topology, infinite-dimensional Lie algebras, and applications (Advances in the Mathematical Sciences) Volume 194, American Mathematical Society, 1999, pp. 81-112 | Article | MR 1729360 | Zbl 1076.14512

[5] Kac, Victor G. Infinite-dimensional Lie algebras, Cambridge University Press, 1990, xxii+400 pages | Article | MR 1104219 | Zbl 0716.17022

[6] Muthiah, Dinakar On Iwahori-Hecke algebras for p-adic loop groups: double coset basis and Bruhat order, Am. J. Math., Volume 140 (2018) no. 1, pp. 221-244 | Article | MR 3749194 | Zbl 1390.22019