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.

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

Halverson, Tom ^{1};
Jacobson, Theodore N. ^{2}

@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}, pages = {509--538}, publisher = {MathOA foundation}, volume = {3}, number = {2}, year = {2020}, doi = {10.5802/alco.102}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.102/} }

TY - JOUR AU - Halverson, Tom AU - Jacobson, Theodore N. TI - Set-partition tableaux and representations of diagram algebras JO - Algebraic Combinatorics PY - 2020 SP - 509 EP - 538 VL - 3 IS - 2 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.102/ DO - 10.5802/alco.102 LA - en ID - ALCO_2020__3_2_509_0 ER -

%0 Journal Article %A Halverson, Tom %A Jacobson, Theodore N. %T Set-partition tableaux and representations of diagram algebras %J Algebraic Combinatorics %D 2020 %P 509-538 %V 3 %N 2 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.102/ %R 10.5802/alco.102 %G en %F 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/articles/10.5802/alco.102/

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

[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 | DOI | MR

[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 | DOI | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

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

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

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

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

[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

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

[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 | DOI | MR | Zbl

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

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

[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 | DOI | Zbl

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

[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 | DOI | MR | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

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

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