Set-partition tableaux and representations of diagram algebras

Halverson, Tom; Jacobson, Theodore N.

Algebraic Combinatorics, Tome 3 (2020) no. 2, pp. 509-538.

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.

Classification : 05E10, 05E18, 20C15
Mots-clés : Set partitions, tableaux, partition algebra, symmetric group, Brauer algebra, Temperley–Lieb algebra, Motzkin algebra, Rook monoid.

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

[2] Benkart, Georgia; Halverson, Tom 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] Benkart, Georgia; Halverson, Tom 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] Benkart, Georgia; Halverson, Tom; Harman, Nate 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] Brauer, Richard On algebras which are connected with the semisimple continuous groups, Ann. Math. (2), Volume 38 (1937), pp. 857-872 | DOI | MR | Zbl

[6] Doran IV, William F.; Wales, David B. The partition algebra revisited, J. Algebra, Volume 231 (2000) no. 1, pp. 265-330 | DOI | MR | Zbl

[7] Doran IV, William F.; Wales, David B.; Hanlon, Philip J. On the semisimplicity of the Brauer centralizer algebras, J. Algebra, Volume 211 (1999) no. 2, pp. 647-685 | DOI | MR | Zbl

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

[9] Farina, John; Halverson, Tom 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] Flath, Daniel; Halverson, Tom; Herbig, Kathryn The planar rook algebra and Pascal’s triangle, Enseign. Math. (2), Volume 55 (2009) no. 1-2, pp. 77-92 | DOI | MR | Zbl

[11] Garsia, Adriano M.; McLarnan, Timothy J. 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] Halverson, Tom Characters of the partition algebras, J. Algebra, Volume 238 (2001) no. 2, pp. 502-533 | DOI | MR | Zbl

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

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

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

[16] Halverson, Tom; Ram, Arun 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] Halverson, Tom; Ram, Arun Partition algebras, Eur. J. Comb., Volume 26 (2005) no. 6, pp. 869-921 | DOI | MR | Zbl

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

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

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

[21] James, Gordon; Kerber, Adalbert 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] Jones, Vaughan F. R. 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] Knuth, Donald E. Two notes on notation, Am. Math. Mon., Volume 99 (1992) no. 5, pp. 403-422 | DOI | MR | Zbl

[24] Kudryavtseva, Ganna; Mazorchuk, Volodymyr 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] Martin, Paul P. Representations of graph Temperley–Lieb Algebras, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 3, pp. 485-503 | DOI | MR | Zbl

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

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

[29] Martin, Paul P. 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] Martin, Paul P.; Mazorchuk, Volodymyr On the representation theory of partial Brauer algebras, Q. J. Math., Volume 65 (2014) no. 1, pp. 225-247 | DOI | MR | Zbl

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

[32] Martin, Paul P.; Saleur, Hubert 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] Nazarov, Maxim Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra, Volume 182 (1996) no. 3, pp. 664-693 | DOI | MR | Zbl

[34] Orellana, Rosa; Zabrocki, Mike Symmetric group characters as symmetric functions (2016) (https://arxiv.org/abs/1605.06672)

[35] Ram, Arum 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] Sagan, Bruce The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Grad. Texts Math., 203, Springer, New York, 2001 | Zbl

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

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

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

[40] Temperley, H.N.V.; Lieb, Eliott H. 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] Westbury, Bruce The representation theory of the Temperley–Lieb algebras, Math. Z., Volume 219 (1995) no. 4, pp. 539-565 | DOI | MR | Zbl