Geodesic
Gram matrices and Stirling numbers of a class of diagram algebras,~I
Algebra and discrete mathematics, Tome 25 (2018) no. 1, pp. 73-97.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we introduce Gram matrices for the signed partition algebras, the algebra of $\mathbb{Z}_2$-relations and the partition algebras. The nondegeneracy and symmetic nature of these Gram matrices are establised. Also, $(s_1, s_2, r_1, r_2, p_1, p_2)$-Stirling numbers of the second kind for the signed partition algebras, the algebra of $\mathbb{Z}_2$-relations are introduced and their identities are established. Stirling numbers of the second kind for the partition algebras are introduced and their identities are established. 
@article{ADM_2018_25_1_a6,
     author = {N. Karimilla Bi and M. Parvathi},
     title = {Gram matrices and {Stirling} numbers of a class of diagram {algebras,~I}},
     journal = {Algebra and discrete mathematics},
     pages = {73--97},
     publisher = {mathdoc},
     volume = {25},
     number = {1},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2018_25_1_a6/}
}
TY  - JOUR
AU  - N. Karimilla Bi
AU  - M. Parvathi
TI  - Gram matrices and Stirling numbers of a class of diagram algebras,~I
JO  - Algebra and discrete mathematics
PY  - 2018
SP  - 73
EP  - 97
VL  - 25
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2018_25_1_a6/
LA  - en
ID  - ADM_2018_25_1_a6
ER  - 
%0 Journal Article
%A N. Karimilla Bi
%A M. Parvathi
%T Gram matrices and Stirling numbers of a class of diagram algebras,~I
%J Algebra and discrete mathematics
%D 2018
%P 73-97
%V 25
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2018_25_1_a6/
%G en
%F ADM_2018_25_1_a6
N. Karimilla Bi; M. Parvathi. Gram matrices and Stirling numbers of a class of diagram algebras,~I. Algebra and discrete mathematics, Tome 25 (2018) no. 1, pp. 73-97. http://geodesic.mathdoc.fr/item/ADM_2018_25_1_a6/

[1] J. J. Graham and G. I. Lehrer, “Cellular Algebras”, Inventiones Mathematicae, 123 (1996), 1–34 | DOI | MR | Zbl

[2] Arun Ram, Tom Halverson, “The partition algebras”, European Journal of electronics, 26 (2005), 869–921 | MR | Zbl

[3] V. F. R. Jones, “The Potts model and the symmetric group”, Subfactors (Kyuzeso), World Sci. Publ., River Edge, NJ, 1993, 259–267 | MR

[4] S. König and C. Xi, “When is a cellular algebra quasi-hereditary?”, Math. Ann., 315:2 (1999), 281–293 | DOI | MR | Zbl

[5] N. Karimilla Bi, Cellularity of a larger class of diagram algebras, accepted for publication in Kyunpook Mathematical journal, arXiv: 1506.02780 | MR

[6] M. Parvathi, “Signed partition algebras”, Comm. Algebra, 32:5 (2004), 1865–1880 | DOI | MR | Zbl

[7] P. Martin and H. Saleur, “Algebras in higher-dimensional statistical mechanics—the exceptional partition (mean field) algebras”, Lett. Math. Phys., 30:3 (1994), 179–185 | DOI | MR | Zbl

[8] P. P. Martin, “Representations of graph Temperley-Lieb algebras”, Publ. Res. Inst. Math. Sci., 26:3 (1990), 485–503 | DOI | MR | Zbl

[9] P. Martin, “Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction”, J. Knot Theory Ramifications, 3:1 (1994), 51–82 | DOI | MR | Zbl

[10] P. Martin, “The structure of the partition algebras”, J. Algebra, 183:2 (1996), 319–358 | DOI | MR | Zbl

[11] P. Martin, “The partition algebra and the Potts model transfer matrix spectrum in high dimension”, J. Phys. A, 33 (2000), 3669–3695 | DOI | MR | Zbl

[12] P. P. Martin, “Potts models and related problems in statistical mechanics”, Advances in Statistical Mechanics, 5, World Sientific Publishing Co. Inc., Teaneck NJ, 1991 | MR | Zbl

[13] M. Parvathi, C. Selvaraj, “Signed Brauer's algebra as centralizer algebras”, Comm. in Algebra, 27:12 (1999), 5985–5998 | DOI | MR | Zbl

[14] Richard P. Stanley, Enumerative combinatorics, v. I, Cambridge studies in Advanced mathematics, 49 | MR

[15] V. Kodiyalam, R. Srinivasan and V. S. Sunder, “The algebra of $G$-relations”, Proc. Indian Acad. Sci. Math. Sci., 110:3 (2000), 263–292 | DOI | MR | Zbl

[16] H. Wenzl, “Representations of Hecke algebras of type $A_n$ and subfactors”, Invent. Math., 92 (1988), 349–383 | DOI | MR | Zbl

[17] C. Xi, “Partition algebras are cellular”, Compositio Math., 119:1 (1999), 99–109 | MR | Zbl