Representation stability for sequences of 0-Hecke modules

We define a new category analogous to ${\bf FI}$ for the $0$-Hecke algebra $H_n(0)$ called the $0$-Hecke category, $\mathcal{H}$, indexing sequences of representations of $H_n(0)$ as $n$ varies under suitable compatibility conditions. We establish a new type of representation stability in this setting and prove it is implied by being a finitely generated $\mathcal{H}$-module. We then provide examples of $\mathcal{H}$-modules and discuss further desirable properties these modules possess.

[1]  Jan Draisma,et al.  Noetherianity up to Symmetry , 2013, 1310.1705.

[2]  P. Norton,et al.  0-Hecke algebras , 1979 .

[3]  Sarah Mason,et al.  Quasisymmetric Schur functions , 2011, J. Comb. Theory, Ser. A.

[4]  The representation theory of Brauer categories I: triangular categories , 2020, 2006.04328.

[5]  Thomas Lam,et al.  P-partition products and fundamental quasi-symmetric function positivity , 2008, Adv. Appl. Math..

[6]  Robert P. Laudone.  Syzygies of secant ideals of Pl\"ucker-embedded Grassmannians are generated in bounded degree , 2018, 1803.04259.

[7]  Steven V. Sam,et al.  Noetherianity of some degree two twisted commutative algebras , 2015, 1501.06925.

[8]  Jia Huang,et al.  0-Hecke Algebra Action on the Stanley-Reisner Ring of the Boolean Algebra , 2013, 1306.1931.

[9]  Andrew Snowden,et al.  The Representation Theory of the Increasing Monoid , 2018, Memoirs of the American Mathematical Society.

[10]  D. Quillen.  On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field , 1972 .

[11]  Steven V. Sam,et al.  Introduction to twisted commutative algebras , 2012, 1209.5122.

[12]  Brendon Rhoades,et al.  Ordered set partitions and the 0-Hecke algebra , 2016, 1611.01251.

[13]  Sarah K. Mason.  Recent Trends in Quasisymmetric Functions , 2018, Association for Women in Mathematics Series.

[14]  G. Whaples,et al.  Hecke operators in cohomology of groups , 1970 .

[15]  Steven V. Sam,et al.  STABILITY PATTERNS IN REPRESENTATION THEORY , 2013, Forum of Mathematics, Sigma.

[16]  Daniel Krob,et al.  Noncommutative Symmetric Functions Iv: Quantum Linear Groups and Hecke Algebras at q = 0 , 1997 .

[17]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[18]  Steven V. Sam,et al.  Gröbner methods for representations of combinatorial categories , 2014, 1409.1670.

[19]  Richard Brauer,et al.  On Algebras Which are Connected with the Semisimple Continuous Groups , 1937 .

[20]  W. Lück.  Transformation groups and algebraic K-theory , 1989 .

[21]  O. Randal-Williams,et al.  Homological stability for automorphism groups , 2014, 1409.3541.

[22]  Paul Martin,et al.  The partition algebra and the Potts model transfer matrix spectrum in high dimensions , 2000 .

[23]  W. Geigle.  The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences , 1985 .

[24]  ANDREW SNOWDEN.  SYZYGIES OF SEGRE EMBEDDINGS AND ∆-MODULES , 2011 .

[25]  Braid groups and Iwahori-Hecke algebras , 2005, math/0505064.

[26]  Jordan S. Ellenberg,et al.  FI-modules and stability for representations of symmetric groups , 2012, 1204.4533.

[27]  J. Ellenberg,et al.  FI-modules over Noetherian rings , 2012, 1210.1854.

[28]  V. Turaev.  OPERATOR INVARIANTS OF TANGLES, AND R-MATRICES , 1990 .

[29]  Wee Liang Gan,et al.  An inductive machinery for representations of categories with shift functors , 2016, Transactions of the American Mathematical Society.

[30]  Min Ho Lee.  Hecke operators on cohomology , 2009 .

[31]  W. Hager,et al.  and s , 2019, Shallow Water Hydraulics.

[32]  Bitraces for GLn(Fq) and the Iwahori-Hecke algebra of type An − 1 , 1999 .

[33]  Philip Tosteson.  Stability in the homology of Deligne–Mumford compactifications , 2018, Compositio Mathematica.

[34]  Andrew Snowden,et al.  Syzygies of Segre embeddings and $\Delta$-modules , 2010, 1006.5248.

[35]  J. Draisma,et al.  Plücker varieties and higher secants of Sato's Grassmannian , 2014, 1402.1667.

[36]  Steven V. Sam,et al.  Stability in the homology of unipotent groups , 2017, Algebra & Number Theory.

[37]  Joe Harris,et al.  Representation Theory: A First Course , 1991 .

[38]  Peter Patzt.  Representation stability for diagram algebras , 2020, Journal of Algebra.

[39]  Victor Reiner,et al.  Hopf Algebras in Combinatorics , 2014, 1409.8356.

[40]  K. Koike.  On the decomposition of tensor products of the representations of the classical groups: By means of the universal characters , 1989 .

[41]  Steven V. Sam,et al.  Ideals of bounded rank symmetric tensors are generated in bounded degree , 2015, 1608.01722.