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A Demazure Character Formula for the Product Monomial Crystal
Gibson, Joel1
1 School of Mathematics and Statistics University of Sydney NSW 2006, Australia
Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 301-327.

The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra 𝔤, and depends on a collection of parameters R. We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. This character formula shows that the product monomial crystal is the crystal of a generalised Demazure module, as defined by Lakshmibai, Littelmann and Magyar. In type A, we show the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram depending on R.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.156
Classification: 05E10
Keywords: Monomial crystal, generalised schur module, Demazure crystal.

Gibson, Joel 1

1 School of Mathematics and Statistics University of Sydney NSW 2006, Australia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Gibson, Joel. A Demazure Character Formula for the Product Monomial Crystal. Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 301-327. doi : 10.5802/alco.156. https://alco.centre-mersenne.org/articles/10.5802/alco.156/

[1] Braden, Tom; Licata, Anthony; Proudfoot, Nicholas; Webster, Ben Quantizations of Conical Symplectic Resolutions II: Category đť’Ş and Symplectic Duality (2014) (https://arxiv.org/abs/1407.0964) | Zbl

[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 | DOI | MR | Zbl

[3] Bump, Daniel; Schilling, Anne Crystal bases. Representations and combinatorics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, xii+279 pages | DOI | MR | Zbl

[4] Demazure, Michel Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4), Volume 7 (1974) no. 1, pp. 53-88 | DOI | Numdam | MR | Zbl

[5] Hernandez, David; Nakajima, Hiraku Level 0 monomial crystals, Nagoya Math. J., Volume 184 (2006), pp. 85-153 | DOI | MR | Zbl

[6] Joseph, Anthony A decomposition theorem for Demazure crystals, J. Algebra, Volume 265 (2003) no. 2, pp. 562-578 | DOI | MR | Zbl

[7] Kamnitzer, Joel; Tingley, Peter; Webster, Ben; Weekes, Alex; Yacobi, Oded Highest weights for truncated shifted Yangians and product monomial crystals, J. Comb. Algebra, Volume 3 (2019) no. 3, pp. 237-303 | DOI | MR | Zbl

[8] Kamnitzer, Joel; Tingley, Peter; Webster, Ben; Weekes, Alex; Yacobi, Oded On category đť’Ş for affine Grassmannian slices and categorified tensor products, Proc. Lond. Math. Soc. (3), Volume 119 (2019) no. 5, pp. 1179-1233 | DOI | MR | Zbl

[9] Kamnitzer, Joel; Webster, Ben; Weekes, Alex; Yacobi, Oded Yangians and quantizations of slices in the affine Grassmannian, Algebra Number Theory, Volume 8 (2014) no. 4, pp. 857-893 | DOI | MR | Zbl

[10] Kashiwara, Masaki Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys., Volume 133 (1990) no. 2, pp. 249-260 | DOI | MR | Zbl

[11] Kashiwara, Masaki On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J., Volume 63 (1991) no. 2, pp. 465-516 | DOI | MR | Zbl

[12] Kashiwara, Masaki The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J., Volume 71 (1993) no. 3, pp. 839-858 | DOI | MR | Zbl

[13] Kashiwara, Masaki Crystal bases of modified quantized enveloping algebra, Duke Math. J., Volume 73 (1994) no. 2, pp. 383-413 | DOI | MR | Zbl

[14] Kashiwara, Masaki Realizations of Crystals (2002) (https://arxiv.org/abs/math/0202268) | Zbl

[15] Khovanov, Mikhail; Lauda, Aaron D. A diagrammatic approach to categorification of quantum groups. I, Represent. Theory, Volume 13 (2009), pp. 309-347 | DOI | MR | Zbl

[16] Khovanov, Mikhail; Lauda, Aaron D. A diagrammatic approach to categorification of quantum groups. II, Trans. Amer. Math. Soc., Volume 363 (2011) no. 5, pp. 2685-2700 | DOI | MR | Zbl

[17] Lakshmibai, Venkatramani; Littelmann, Peter; Magyar, Peter Standard monomial theory for Bott–Samelson varieties, Compositio Math., Volume 130 (2002) no. 3, pp. 293-318 | DOI | MR | Zbl

[18] Magyar, Peter Borel-Weil theorem for configuration varieties and Schur modules, Adv. Math., Volume 134 (1998) no. 2, pp. 328-366 | DOI | MR | Zbl

[19] Magyar, Peter Schubert polynomials and Bott–Samelson varieties, Comment. Math. Helv., Volume 73 (1998) no. 4, pp. 603-636 | DOI | MR | Zbl

[20] Mirković, Ivan; Vilonen, Kari Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2), Volume 166 (2007) no. 1, pp. 95-143 | DOI | MR | Zbl

[21] Reiner, Victor; Shimozono, Mark Key polynomials and a flagged Littlewood–Richardson rule, J. Combin. Theory Ser. A, Volume 70 (1995) no. 1, pp. 107-143 | DOI | MR | Zbl

[22] Reiner, Victor; Shimozono, Mark Percentage-avoiding, northwest shapes and peelable tableaux, J. Combin. Theory Ser. A, Volume 82 (1998) no. 1, pp. 1-73 | DOI | MR | Zbl

[23] Reiner, Victor; Shimozono, Mark Flagged Weyl modules for two column shapes, J. Pure Appl. Algebra, Volume 141 (1999) no. 1, pp. 59-100 | DOI | MR | Zbl

[24] Rouquier, Raphael 2-Kac-Moody Algebras (2008) (https://arxiv.org/abs/0812.5023) | Zbl

[25] Webster, Ben; Weekes, Alex; Yacobi, Oded A Quantum Mirković–Vybornov Isomorphism (2017) (https://arxiv.org/abs/1706.03841) | Zbl

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