The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley–Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We construct the irreducible modules of these algebras in three isomorphic ways: as the span of diagrams in a quotient of the left regular representation; as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation; and on a basis indexed by set-partition tableaux such that diagrams in the algebra act combinatorially on tableaux. The second representation is analogous to the Gelfand model and the third is a generalization of Young’s natural representation of the symmetric group on standard tableaux. The methods of this paper work uniformly for the partition algebra and its diagram subalgebras. As an application, we express the characters of each of these algebras as nonnegative integer combinations of symmetric group characters whose coefficients count fixed points under conjugation.

Revised: 2019-04-26

Accepted: 2019-11-23

Published online: 2020-04-01

Classification: 05E10, 05E18, 20C15

Keywords: Set partitions, tableaux, partition algebra, symmetric group, Brauer algebra, Temperley–Lieb algebra, Motzkin algebra, Rook monoid.

@article{ALCO_2020__3_2_509_0, author = {Halverson, Tom and Jacobson, Theodore N.}, title = {Set-partition tableaux and representations of diagram algebras}, journal = {Algebraic Combinatorics}, publisher = {MathOA foundation}, volume = {3}, number = {2}, year = {2020}, pages = {509-538}, doi = {10.5802/alco.102}, language = {en}, url={alco.centre-mersenne.org/item/ALCO_2020__3_2_509_0/} }

Halverson, Tom; Jacobson, Theodore N. Set-partition tableaux and representations of diagram algebras. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 509-538. doi : 10.5802/alco.102. https://alco.centre-mersenne.org/item/ALCO_2020__3_2_509_0/

[1] Motzkin algebras, Eur. J. Comb., Volume 36 (2014), pp. 473-502 | Article | MR 3131911 | Zbl 1284.05333

[2] Partition algebras and the invariant theory of the symmetric group, Recent Trends in Algebraic Combinatorics (Barcelo, H.; Karaali, G.; Orellana, R., eds.) (Association for Women in Mathematics Series) Volume 16, Springer, 2019, pp. 1-41 | Article | MR 3969570

[3] Partition algebras ${P}_{k}\left(n\right)$ with $2k>n$ and the fundamental theorems of invariant theory for the symmetric group ${S}_{n}$, J. Lond. Math. Soc., Volume 99 (2019) no. 1, pp. 194-224 | Article

[4] Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups, J. Algebr. Comb., Volume 46 (2017) no. 1, pp. 77-108 | Article | MR 3666413 | Zbl 1368.05156

[5] On algebras which are connected with the semisimple continuous groups, Ann. Math. (2), Volume 38 (1937), pp. 857-872 | Article | MR 1503378 | Zbl 0017.39105

[6] The partition algebra revisited, J. Algebra, Volume 231 (2000) no. 1, pp. 265-330 | Article | MR 1779601 | Zbl 0974.20013

[7] On the semisimplicity of the Brauer centralizer algebras, J. Algebra, Volume 211 (1999) no. 2, pp. 647-685 | Article | MR 1666664 | Zbl 0944.16002

[8] A seminormal form for partition algebras, J. Comb. Theory, Ser. A, Volume 120 (2013) no. 7, pp. 1737-1785 | Article | MR 3092697 | Zbl 1314.05219

[9] Character orthogonality for the partition algebra and fixed points of permutations, Adv. Appl. Math., Volume 31 (2003) no. 1, pp. 113-131 | Article | MR 1985823 | Zbl 1020.05073

[10] The planar rook algebra and Pascal’s triangle, Enseign. Math. (2), Volume 55 (2009) no. 1-2, pp. 77-92 | Article | MR 2541502 | Zbl 1209.20004

[11] Relations between Young’s natural and the Kazhdan–Lusztig representations of ${S}_{n}$, Adv. Math., Volume 69 (1988) no. 1, pp. 32-92 | Article

[12] Characters of the partition algebras, J. Algebra, Volume 238 (2001) no. 2, pp. 502-533 | Article | MR 1823772 | Zbl 1083.20010

[13] Representations of the $q$-rook monoid, J. Algebra, Volume 273 (2004) no. 1, pp. 227-251 | Article | MR 2032458

[14] Representations of the rook-Brauer algebra, Commun. Algebra, Volume 42 (2014) no. 1, pp. 423-443 | Article | MR 3169580 | Zbl 1291.05215

[15] Commuting families in Hecke and Temperley–Lieb algebras, Nagoya Math. J., Volume 195 (2009), pp. 125-152 | Article | MR 2552957 | Zbl 1217.20002

[16] Characters of algebras containing a Jones basic construction: The Temperley–Lieb, Okada, Brauer, and Birman–Wenzl algebras, Adv. Math., Volume 116 (1995) no. 2, pp. 263-321 | Article | MR 1363766 | Zbl 0856.16038

[17] Partition algebras, Eur. J. Comb., Volume 26 (2005) no. 6, pp. 869-921 | Article | MR 2143201 | Zbl 1112.20010

[18] Gelfand models for diagram algebras, J. Algebr. Comb., Volume 41 (2015) no. 2, pp. 229-255 | Article | MR 3306071 | Zbl 1308.05107

[19] On the decomposition of Brauer’s centralizer algebras, J. Algebra, Volume 121 (1989) no. 2, pp. 409-445 | Article | MR 992775 | Zbl 0695.20026

[20] The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, Volume 682, Springer-Verlag, 1978 | Article | MR 513828

[21] The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Volume 16, Addison-Wesley Publishing Co., Reading, Mass., 1981, xxviii+510 pages | MR 644144 | Zbl 0491.20010

[22] The Potts model and the symmetric group, Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras (Kyuzeso, 1993), World Scientific, 1994, pp. 259-267 | Zbl 0938.20505

[23] Two notes on notation, Am. Math. Mon., Volume 99 (1992) no. 5, pp. 403-422 | Article | MR 1163629 | Zbl 0785.05014

[24] Combinatorial Gelfand models for some semigroups and $q$-rook monoid algebras, Proc. Edinb. Math. Soc., II. Ser., Volume 52 (2009) no. 3, pp. 707-718 | Article | MR 2546640 | Zbl 1201.20066

[25] Representations of graph Temperley–Lieb Algebras, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 3, pp. 485-503 | Article | MR 1068862 | Zbl 0718.17025

[26] Potts Models and Related Problems in Statistical Mechanics, Series on advances in statistical mechanics, Volume 5, World Scientific, 1991 | MR 1103994 | Zbl 0734.17012

[27] Temperley–Lieb algebras for non-planar statistical mechanics — the partition algebra construction, J. Knot Theory Ramifications, Volume 03 (1994) no. 01, pp. 51-82 | Article | Zbl 0804.16002

[28] The structure of the partition algebras, J. Algebra, Volume 183 (1996), pp. 319-358 | Article | MR 1399030 | Zbl 0863.20009

[29] The partition algebra and the Potts model transfer matrix spectrum in high dimensions, J. Phys. A, Math. Gen., Volume 33 (2000) no. 19, pp. 3669-3695 | Article | Zbl 0951.82006

[30] On the representation theory of partial Brauer algebras, Q. J. Math., Volume 65 (2014) no. 1, pp. 225-247 | Article | MR 3179659 | Zbl 1354.16016

[31] On an algebraic approach to higher-dimensional statistical mechanics, Comm. Math. Phys., Volume 158 (1993) no. 1, pp. 155-190 | Article | MR 1243720 | Zbl 0784.05056

[32] Algebras in higher-dimensional statistical mechanics — the exceptional partition (mean field) algebras, Lett. Math. Phys., Volume 30 (1994) no. 3, pp. 179-185 | Article | MR 1266999 | Zbl 0799.16004

[33] Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra, Volume 182 (1996) no. 3, pp. 664-693 | Article | MR 1398116 | Zbl 0868.20012

[34] Symmetric group characters as symmetric functions (2016) (https://arxiv.org/abs/1605.06672)

[35] Skew shape representations are irreducible, Combinatorial and geometric representation theory (Seoul, 2001) (Contemp. Math.) Volume 325, Amer. Math. Soc., Providence, RI, 2003, pp. 161-189 | Article | MR 1988991 | Zbl 1061.20006

[36] The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Grad. Texts Math., Volume 203, Springer, New York, 2001 | Zbl 0964.05070

[37] On a partition identity, J. Comb. Theory, Ser. A, Volume 36 (1984) no. 2, pp. 249-252 | Article | MR 734984 | Zbl 0532.05003

[38] The On-Line Encyclopedia of Integer Sequences (Published electronically at http://oeis.org) | MR 3822822 | Zbl 06989892

[39] Representations of the rook monoid, J. Algebra, Volume 256 (2002) no. 2, pp. 309-342 | Article | MR 1939108 | Zbl 1034.20056

[40] Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. R. Soc. Lond., Ser. A, Volume 322 (1971) no. 1549, pp. 251-280 | Article | MR 0498284 | Zbl 0211.56703

[41] The representation theory of the Temperley–Lieb algebras, Math. Z., Volume 219 (1995) no. 4, pp. 539-565 | Article | MR 1343661 | Zbl 0840.16008