Browse issues
no. 4
Equivariant Kazhdan–Lusztig polynomials of q-niform matroids
Proudfoot, Nicholas1
1 University of Oregon Department of Mathematics Eugene OR 97403, USA
Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 613-619.

We study q-analogues of uniform matroids, which we call q-niform matroids. While uniform matroids admit actions of symmetric groups, q-niform matroids admit actions of finite general linear groups. We show that the equivariant Kazhdan–Lusztig polynomial of a q-niform matroid is the unipotent q-analogue of the equivariant Kazhdan–Lusztig polynomial of the corresponding uniform matroid, thus providing evidence for the positivity conjecture for equivariant Kazhdan–Lusztig polynomials.

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/alco.59
Classification: 05B35, 20C33
Keywords: Kazhdan–Lusztig polynomial, matroid, unipotent representation

Proudfoot, Nicholas 1

1 University of Oregon Department of Mathematics Eugene OR 97403, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2019__2_4_613_0,
     author = {Proudfoot, Nicholas},
     title = {Equivariant {Kazhdan{\textendash}Lusztig} polynomials of $q$-niform matroids},
     journal = {Algebraic Combinatorics},
     pages = {613--619},
     publisher = {MathOA foundation},
     volume = {2},
     number = {4},
     year = {2019},
     doi = {10.5802/alco.59},
     zbl = {1417.05024},
     mrnumber = {3997514},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.59/}
}
TY  - JOUR
AU  - Proudfoot, Nicholas
TI  - Equivariant Kazhdan–Lusztig polynomials of $q$-niform matroids
JO  - Algebraic Combinatorics
PY  - 2019
SP  - 613
EP  - 619
VL  - 2
IS  - 4
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.59/
DO  - 10.5802/alco.59
LA  - en
ID  - ALCO_2019__2_4_613_0
ER  - 
%0 Journal Article
%A Proudfoot, Nicholas
%T Equivariant Kazhdan–Lusztig polynomials of $q$-niform matroids
%J Algebraic Combinatorics
%D 2019
%P 613-619
%V 2
%N 4
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.59/
%R 10.5802/alco.59
%G en
%F ALCO_2019__2_4_613_0
Proudfoot, Nicholas. Equivariant Kazhdan–Lusztig polynomials of $q$-niform matroids. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 613-619. doi : 10.5802/alco.59. https://alco.centre-mersenne.org/articles/10.5802/alco.59/

[1] Andrews, Scott The unipotent modules of GL n (𝔽 q ) via tableaux, J. Algebraic Combin., Volume 47 (2018) no. 1, pp. 1-15 | DOI | MR

[2] Benson, Clark T.; Curtis, Charles W. On the degrees and rationality of certain characters of finite Chevalley groups, Trans. Amer. Math. Soc., Volume 165 (1972), pp. 251-273 | DOI | MR | Zbl

[3] Church, Thomas; Ellenberg, Jordan S.; Farb, Benson FI-modules and stability for representations of symmetric groups, Duke Math. J., Volume 164 (2015) no. 9, pp. 1833-1910 | DOI | MR | Zbl

[4] Curtis, Charles W. Reduction theorems for characters of finite groups of Lie type, J. Math. Soc. Japan, Volume 27 (1975) no. 4, pp. 666-688 | DOI | MR | Zbl

[5] Dipper, Richard; James, Gordon On Specht modules for general linear groups, J. Algebra, Volume 275 (2004) no. 1, pp. 106-142 | DOI | MR | Zbl

[6] Dudas, Olivier Lectures on modular Deligne-Lusztig theory, Local Representation Theory and Simple Groups (Series of Lectures in Mathematics), Volume 29, European Mathematical Society, 2018, pp. 107-177 | DOI | MR | Zbl

[7] Elias, Ben; Proudfoot, Nicholas; Wakefield, Max The Kazhdan-Lusztig polynomial of a matroid, Adv. Math., Volume 299 (2016), pp. 36-70 | DOI | MR | Zbl

[8] Gan, Wee Liang; Watterlond, John A representation stability theorem for VI-modules, Algebr. Represent. Theory, Volume 21 (2018) no. 1, pp. 47-60 | MR | Zbl

[9] Gao, Alice L. L.; Lu, Linyuan; Xie, Matthew H. Y.; Yang, Arthur L. B.; Zhang, Philip B. The Kazhdan-Lusztig polynomials of uniform matroids (https://arxiv.org/abs/1806.10852)

[10] Gedeon, Katie; Proudfoot, Nicholas; Young, Benjamin The equivariant Kazhdan–Lusztig polynomial of a matroid, J. Combin. Theory Ser. A, Volume 150 (2017), pp. 267-294 | DOI | MR | Zbl

[11] Gedeon, Katie R. Kazhdan-Lusztig polynomials of thagomizer matroids, Electron. J. Combin., Volume 24 (2017) no. 3, Paper no. P3.12, 10 pages | MR | Zbl

[12] Hameister, Thomas; Rao, Sujit; Simpson, Connor Chow rings of vector space matroids (https://arxiv.org/abs/1802.04241)

[13] Howlett, Robert B.; Lehrer, Gustav I. Representations of generic algebras and finite groups of Lie type, Trans. Amer. Math. Soc., Volume 280 (1983) no. 2, pp. 753-779 | DOI | MR | Zbl

[14] Karn, Trevor K.; Wakefield, Max D. Stirling numbers in braid matroid Kazhdan-Lusztig polynomials (https://arxiv.org/abs/1802.00849) | Zbl

[15] Lu, Linyuan; Xie, Matthew H. Y.; Yang, Arthur L. B. Kazhdan-Lusztig polynomials of fan matroids, wheel matroids and whirl matroids (https://arxiv.org/abs/1802.03711)

[16] Lusztig, George Coxeter orbits and eigenspaces of Frobenius, Invent. Math., Volume 38 (1976) no. 2, pp. 101-159 | DOI | MR | Zbl

[17] Orlik, Peter; Solomon, Louis Combinatorics and topology of complements of hyperplanes, Invent. Math., Volume 56 (1980) no. 2, pp. 167-189 | DOI | MR | Zbl

[18] Proudfoot, Nicholas; Young, Ben Configuration spaces, FS op -modules, and Kazhdan-Lusztig polynomials of braid matroids, New York J. Math., Volume 23 (2017), pp. 813-832 | MR | Zbl

Cited by Sources: