@book {Tits, AUTHOR = {Tits, Jacques}, TITLE = {Buildings of spherical type and finite {BN}-pairs}, SERIES = {Lecture Notes in Mathematics}, VOLUME = {Vol. 386}, PUBLISHER = {Springer-Verlag, Berlin-New York}, YEAR = {1974}, PAGES = {x+299}, MRCLASS = {20G15 (50A20)}, MRNUMBER = {470099}, MRREVIEWER = {Bruce\ Cooperstein}, } @book {Hatcher, AUTHOR = {Hatcher, Allen}, TITLE = {Algebraic topology}, PUBLISHER = {Cambridge University Press, Cambridge}, YEAR = {2002}, PAGES = {xii+544}, ISBN = {0-521-79160-X; 0-521-79540-0}, MRCLASS = {55-01 (55-00)}, MRNUMBER = {1867354}, MRREVIEWER = {Donald\ W.\ Kahn}, } @book {Macdonald, AUTHOR = {Macdonald, I. G.}, TITLE = {Symmetric functions and {H}all polynomials}, SERIES = {Oxford Classic Texts in the Physical Sciences}, EDITION = {Second}, NOTE = {With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition [ MR1354144]}, PUBLISHER = {The Clarendon Press, Oxford University Press, New York}, YEAR = {2015}, PAGES = {xii+475}, ISBN = {978-0-19-873912-8}, MRCLASS = {05E05 (01A75 05-02 20C30 20C33 20K01 33C80 33D80)}, MRNUMBER = {3443860}, } @book {Stanley-EC2, AUTHOR = {Stanley, Richard P.}, TITLE = {Enumerative combinatorics. {V}ol. 2}, SERIES = {Cambridge Studies in Advanced Mathematics}, VOLUME = {62}, NOTE = {With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin}, PUBLISHER = {Cambridge University Press, Cambridge}, YEAR = {1999}, PAGES = {xii+581}, ISBN = {0-521-56069-1; 0-521-78987-7}, MRCLASS = {05A15 (05-02 05E05 05E10 68R05)}, MRNUMBER = {1676282}, MRREVIEWER = {Ira Gessel}, DOI = {10.1017/CBO9780511609589}, URL = {https://doi-org.ezp1.lib.umn.edu/10.1017/CBO9780511609589}, } @article{brauner2022eulerian, title={Eulerian representations for real reflection groups}, author={Brauner, Sarah}, journal={Journal of the London Mathematical Society}, volume={105}, number={1}, pages={412--444}, year={2022}, publisher={Wiley Online Library} } @article{aguiar2004peak, title={The peak algebra and the descent algebras of types {B} and {D}}, author={Aguiar, Marcelo and Bergeron, Nantel and Nyman, Kathryn}, journal={Transactions of the American Mathematical Society}, volume={356}, number={7}, pages={2781--2824}, year={2004} } @article {aguiar2006peak, AUTHOR = {Aguiar, Marcelo and Nyman, Kathryn and Orellana, Rosa}, TITLE = {New results on the peak algebra}, JOURNAL = {J. Algebraic Combin.}, FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal}, VOLUME = {23}, YEAR = {2006}, NUMBER = {2}, PAGES = {149--188}, ISSN = {0925-9899}, MRCLASS = {20C30}, MRNUMBER = {2223685}, DOI = {10.1007/s10801-006-6922-8}, URL = {https://doi-org.ezp1.lib.umn.edu/10.1007/s10801-006-6922-8}, } @article{moseley2017equivariant, title={Equivariant cohomology and the {V}archenko--{G}elfand filtration}, author={Moseley, Daniel}, journal={Journal of Algebra}, volume={472}, pages={95--114}, year={2017}, publisher={Elsevier} } @article{xico, title={Orbit configuration spaces, infinitesimal braid relations in homology and equivariant loop spaces.}, author={Xicot{\'e}ncatl Merino, Miguel A}, journal={PhD Thesis--University of Rochester.}, year= {1997} } @article{varchenkogelfand, title={Heaviside functions of a configuration of hyperplanes}, author={Varchenko, A. N. and Gel{'}fand, I. M. }, journal={Funktsional Anal. i Prilozhen}, volume={21}, number={4}, pages={1-18}, year={1987}, } @article{orlik-solomon, title={The {H}essian map in the invariant theory of reflection groups}, author={Orlik, Peter and Solomon, Louis}, journal={Nagoya Mathematical Journal}, volume={109}, pages={1--21}, year={1988}, publisher={Cambridge University Press} } @book{cohen, title={The homology of iterated loop spaces}, author={Cohen, Frederick Ronald and Lada, Thomas Joseph and May, Peter J}, year={2007}, publisher={Springer} } @article{calderbank1994representations, title={Representations of the symmetric group in deformations of the free {L}ie algebra}, author={Calderbank, AR and Hanlon, P and Sundaram, S}, journal={Transactions of the American mathematical Society}, volume={341}, number={1}, pages={315--333}, year={1994} } @article{robbins1971jordan, title={Jordan elements in a free associative algebra, {I}}, author={Robbins, David P}, journal={Journal of Algebra}, volume={19}, number={3}, pages={354--378}, year={1971}, publisher={Elsevier} } @article{kashuba2021free, title={On the free {J}ordan algebras}, author={Kashuba, Iryna and Mathieu, Olivier}, journal={Advances in Mathematics}, volume={383}, pages={107690}, year={2021}, publisher={Elsevier} } @article{robinson1996tree, title={The tree representation of ${S}_{n+1}$}, author={Robinson, Alan and Whitehouse, Sarah}, journal={Journal of Pure and Applied Algebra}, volume={111}, number={1-3}, pages={245--253}, year={1996}, publisher={Elsevier} } @book{loday2012algebraic, title={Algebraic operads}, author={Loday, Jean-Louis and Vallette, Bruno}, volume={346}, year={2012}, publisher={Springer} } @misc{DLMF, key = {DLMF, Release 1.0.26 of 2020-03-15}, title = {NIST Digital Library of Mathematical Functions}, howpublished = {http://dlmf.nist.gov/, Release 1.0.26 of 2020-03-15}, year= {2020}, url = {http://dlmf.nist.gov/}, author = {F.~W.~J. Olver and A.~B. Olde Daalhuis and D.~W. Lozier and B.~I. Schneider and R.~F. Boisvert and C.~W. Clark and B.~R. Mille and, B.~V. Saunders and H.~S. Cohl and M.~A. McClain, eds.} } @book{aguiar2017topics, title={Topics in hyperplane arrangements}, author={Aguiar, Marcelo and Mahajan, Swapneel}, volume={226}, year={2017}, publisher={American Mathematical Soc.} } @book{reiner2014spectra, title={Spectra of symmetrized shuffling operators}, author={Reiner, Victor and Saliola, Franco and Welker, Volkmar}, volume={228}, number={1072}, year={2014}, publisher={American mathematical society} } @incollection {reutenauer, AUTHOR = {Reutenauer, Christophe}, TITLE = {Free {L}ie algebras}, BOOKTITLE = {Handbook of algebra, {V}ol. 3}, SERIES = {Handb. Algebr.}, VOLUME = {3}, PAGES = {887--903}, PUBLISHER = {Elsevier/North-Holland, Amsterdam}, YEAR = {2003}, ISBN = {0-444-51264-0}, MRCLASS = {17B01}, MRNUMBER = {2035110}, MRREVIEWER = {Hartmut\ Laue}, DOI = {10.1016/S1570-7954(03)80075-X}, URL = {https://doi.org/10.1016/S1570-7954(03)80075-X}, } @article {gerstenhaberschack, AUTHOR = {Gerstenhaber, Murray and Schack, S. D.}, TITLE = {A {H}odge-type decomposition for commutative algebra cohomology}, JOURNAL = {J. Pure Appl. Algebra}, FJOURNAL = {Journal of Pure and Applied Algebra}, VOLUME = {48}, YEAR = {1987}, NUMBER = {3}, PAGES = {229--247}, ISSN = {0022-4049,1873-1376}, MRCLASS = {13D03 (16A62)}, MRNUMBER = {917209}, MRREVIEWER = {Zbigniew\ Marciniak}, DOI = {10.1016/0022-4049(87)90112-5}, URL = {https://doi.org/10.1016/0022-4049(87)90112-5}, } @article {bergeronbergeron, AUTHOR = {Bergeron, Fran\c{c}ois and Bergeron, Nantel}, TITLE = {Orthogonal idempotents in the descent algebra of {$B_n$} and applications}, JOURNAL = {J. Pure Appl. Algebra}, FJOURNAL = {Journal of Pure and Applied Algebra}, VOLUME = {79}, YEAR = {1992}, NUMBER = {2}, PAGES = {109--129}, ISSN = {0022-4049,1873-1376}, MRCLASS = {20F55 (05E10 16E40)}, MRNUMBER = {1163285}, MRREVIEWER = {John\ R.\ Stembridge}, DOI = {10.1016/0022-4049(92)90153-7}, URL = {https://doi.org/10.1016/0022-4049(92)90153-7}, } @article {hanlon, AUTHOR = {Hanlon, Phil}, TITLE = {The action of {$S_n$} on the components of the {H}odge decomposition of {H}ochschild homology}, JOURNAL = {Michigan Math. J.}, FJOURNAL = {Michigan Mathematical Journal}, VOLUME = {37}, YEAR = {1990}, NUMBER = {1}, PAGES = {105--124}, ISSN = {0026-2285,1945-2365}, MRCLASS = {20C30 (05E10 13D03 18G99)}, MRNUMBER = {1042517}, MRREVIEWER = {Ira\ Gessel}, DOI = {10.1307/mmj/1029004069}, URL = {https://doi.org/10.1307/mmj/1029004069}, } @article {thrall, AUTHOR = {Thrall, R. M.}, TITLE = {On symmetrized {K}ronecker powers and the structure of the free {L}ie ring}, JOURNAL = {Amer. J. Math.}, FJOURNAL = {American Journal of Mathematics}, VOLUME = {64}, YEAR = {1942}, PAGES = {371--388}, ISSN = {0002-9327,1080-6377}, MRCLASS = {09.1X}, MRNUMBER = {6149}, MRREVIEWER = {R.\ Brauer}, DOI = {10.2307/2371691}, URL = {https://doi.org/10.2307/2371691}, } @article{moseley, title={Equivariant cohomology and the {V}archenko--{G}elfand filtration}, author={Moseley, Daniel}, journal={Journal of Algebra}, volume={472}, pages={95--114}, year={2017}, publisher={Elsevier} } @article{barcelo1990action, title={On the action of the symmetric group on the free {L}ie algebra and the partition lattice}, author={Barcelo, H{\'e}l{\`e}ne}, journal={Journal of Combinatorial Theory, Series A}, volume={55}, number={1}, pages={93--129}, year={1990}, publisher={Elsevier} } @article{arnold, title={The cohomology ring of the group of dyed braids}, author={Arnol'd, Vladimir, I}, journal={Mat. Zametki}, volume={5}, pages={227--231}, year={1969} } @article{mahajan2001shuffles, title={Shuffles on Coxeter groups}, author={Mahajan, Swapneel}, journal={arXiv preprint math/0108094}, year={2001} } @article{sundaramwelker, title={Group actions on arrangements of linear subspaces and applications to configuration spaces}, author={Sundaram, Sheila and Welker, Volkmar}, journal={Transactions of the American Mathematical Society}, volume={349}, number={4}, pages={1389--1420}, year={1997} } @article{solomon, title={A {M}ackey formula in the group ring of a {C}oxeter group}, author={Solomon, Louis}, journal={Journal of Algebra}, volume={41}, number={2}, pages={255--264}, year={1976}, publisher={Academic Press} } @article{nyman2003peak, title={The peak algebra of the symmetric group}, author={Nyman, Kathryn L}, journal={Journal of Algebraic Combinatorics}, volume={17}, pages={309--322}, year={2003}, publisher={Springer} } @incollection{reutenauer1986theorem, title={Theorem of {P}oincar{\'e}-{B}irkhoff-{W}itt, logarithm and symmetric group representations of degrees equal to {S}tirling numbers}, author={Reutenauer, Christophe}, booktitle={Combinatoire {\'e}num{\'e}rative}, pages={267--284}, year={1986}, publisher={Springer} } @article{garsiareutenauer, title={A decomposition of {S}olomon's descent algebra}, author={Garsia, Adriano M and Reutenauer, Christophe}, journal={Advances in Mathematics}, volume={77}, number={2}, pages={189--262}, year={1989}, publisher={Elsevier} } @article{lodayoperations, title={Op{\'e}rations sur l'homologie cyclique des alg{\`e}bres commutatives}, author={Loday, Jean-Louis}, journal={Inventiones mathematicae}, volume={96}, number={1}, pages={205--230}, year={1989}, publisher={Springer-Verlag} } @article{BBHT, title={A decomposition of the descent algebra of a finite {C}oxeter group}, author={Bergeron, Fran{\c{c}}ois and Bergeron, Nantel and Howlett, Robert B and Taylor, Donald E}, journal={Journal of Algebraic Combinatorics}, volume={1}, number={1}, pages={23--44}, year={1992}, publisher={Springer} } @article{saliolasemigroup, title={The quiver of the semigroup algebra of a left regular band}, author={Saliola, Franco V}, journal={International Journal of Algebra and Computation}, volume={17}, number={8}, pages={1593--1610}, year={2007}, publisher={World Scientific} } @article{saliola2009face, title={The face semigroup algebra of a hyperplane arrangement}, author={Saliola, Franco V}, journal={Canadian Journal of Mathematics}, volume={61}, number={4}, pages={904--929}, year={2009}, publisher={Cambridge University Press} } @article{saliola2012eigenvectors, title={Eigenvectors for a random walk on a left-regular band}, author={Saliola, Franco V}, journal={Advances in Applied Mathematics}, volume={48}, number={2}, pages={306--311}, year={2012}, publisher={Elsevier} } @article{sal2008quiv, title={On the quiver of the descent algebra}, author={Saliola, Franco V}, journal={Journal of Algebra}, volume={320}, number={11}, pages={3866--3894}, year={2008}, publisher={Elsevier} } @article{bidigare, title={Hyperplane arrangement face algebras and their associated {M}arkov chains}, author={Bidigare, Patrick T}, journal={PhD Thesis--University of Michigan}, year={1998} } @book{benson1998representations, title={Representations and cohomology: Volume 1, basic representation theory of finite groups and associative algebras}, author={Benson, David J}, volume={1}, year={1998}, publisher={Cambridge university press} } @misc{oeis, author={Sloane, Neil J. A.}, title={The on-line encyclopedia of integer sequences: A060524}, year={2020}, url={https://oeis.org/A060524}, }