# ALGEBRAIC COMBINATORICS

Hilbert series for twisted commutative algebras
Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 147-172.

Suppose that for each $n\ge 0$ we have a representation ${M}_{n}$ of the symmetric group ${S}_{n}$. Such sequences arise in a wide variety of contexts, and often exhibit uniformity in some way. We prove a number of general results along these lines in this paper: our prototypical theorem states that if $\left\{{M}_{n}\right\}$ can be given a suitable module structure over a twisted commutative algebra then the sequence $\left\{{M}_{n}\right\}$ follows a predictable pattern. We phrase these results precisely in the language of Hilbert series (or Poincaré series, or formal characters) of modules over tca’s.

Received: 2017-07-28
Revised: 2017-11-28
Accepted: 2017-11-28
Published online: 2018-01-29
DOI: https://doi.org/10.5802/alco.9
Classification: 05E05,  13A50
@article{ALCO_2018__1_1_147_0,
author = {Sam, Steven V and Snowden, Andrew},
title = {Hilbert series for twisted commutative algebras},
journal = {Algebraic Combinatorics},
pages = {147--172},
publisher = {MathOA foundation},
volume = {1},
number = {1},
year = {2018},
doi = {10.5802/alco.9},
zbl = {06882338},
mrnumber = {3857163},
language = {en},
url = {alco.centre-mersenne.org/item/ALCO_2018__1_1_147_0/}
}
Sam, Steven V; Snowden, Andrew. Hilbert series for twisted commutative algebras. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 147-172. doi : 10.5802/alco.9. https://alco.centre-mersenne.org/item/ALCO_2018__1_1_147_0/

[1] Church, Thomas; Ellenberg, Jordan S.; Farb, Benson FI-modules: a new approach to stability for ${S}_{n}$-representations, Duke Math. J, Volume 164 (2015), pp. 1833-1910 | Article | MR 3357185 | Zbl 1339.55004

[2] Fulton, William Intersection Theory, Springer-Verlag, Berlin, 1998 | MR 1644323 | Zbl 0541.14005

[3] Fulton, William; Harris, Joe Representation Theory: A First Course, Graduate Texts in Mathematics, Volume 129, Springer–Verlag, New York, 1991 | MR 1153249 | Zbl 0744.22001

[4] Fulton, William; Lang, Serge Riemann-Roch Algebra, Grundlehren der Mathematischen Wissenschaften, Volume 277, Springer–Verlag, New York, 1985 | MR 801033 | Zbl 0579.14011

[5] Gessel, Ira M. Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A, Volume 53 (1990), pp. 257-285 | Article | MR 1041448 | Zbl 0704.05001

[6] Lipshitz, L. The diagonal of a $D$-finite power series is $D$-finite, J. Algebra, Volume 113 (1988), pp. 373-378 | Article | MR 929767 | Zbl 0657.13024

[7] Macdonald, I. G. Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Oxford, 1995 | MR 1354144 | Zbl 0487.20007

[8] Miller, Ezra; Sturmfels, Bernd Combinatorial Commutative Algebra, Graduate Texts in Mathematics, Springer-Verlag, New York, 2005 no. 227 | MR 2110098 | Zbl 1066.13001

[9] Sam, Steven V; Snowden, Andrew GL-equivariant modules over polynomial rings in infinitely many variables II (arXiv:1703.04516v1) | Zbl 06560453

[10] Sam, Steven V; Snowden, Andrew Introduction to twisted commutative algebras (arXiv:1209.5122v1) | Zbl 1388.05190

[11] Sam, Steven V; Snowden, Andrew Stability patterns in representation theory, Forum Math. Sigma, Volume 3 (2015), e11, 108 pages | Article | MR 3376738 | Zbl 1319.05146

[12] Sam, Steven V; Snowden, Andrew GL-equivariant modules over polynomial rings in infinitely many variables, Trans. Amer. Math. Soc, Volume 368 (2016), pp. 1097-1158 | Article | MR 3430359 | Zbl 06560453

[13] Sam, Steven V; Snowden, Andrew Gröbner methods for representations of combinatorial categories, J. Amer. Math. Soc, Volume 30 (2017), pp. 159-203 | Article | MR 3556290 | Zbl 1347.05010

[14] Snowden, Andrew Syzygies of Segre embeddings and $\Delta$-modules, Duke Math. J, Volume 162 (2013), pp. 225-277 | Article | MR 3018955 | Zbl 1279.13024

[15] Stanley, Richard P. Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, Volume 2, Cambridge University Press, Cambridge, 1999 no. 62 (with a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | MR 1676282 | Zbl 0928.05001

[16] Woodcock, Christopher F.; Sharif, Habib On the transcendence of certain series, J. Algebra, Volume 121 (1989), pp. 364-369 | Article | MR 992771 | Zbl 0689.13014