# ALGEBRAIC COMBINATORICS

Traces on diagram algebras II: centralizer algebras of easy groups and new variations of the Young graph
Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 413-436.

In continuation of our recent work [39], we classify the extremal traces on infinite diagram algebras that appear in the context of Schur–Weyl duality for Banica and Speicher’s easy groups [3]. We show that the branching graphs of these algebras describe walks on new variations of the Young graph which in turn explicate curious ways of growing Young diagrams. As a consequence, we prove that the extremal traces on generic rook-Brauer algebras are always extensions of extremal traces on the group algebra $ℂ\left[{S}_{\infty }\right]$ of the infinite symmetric group. Moreover, we conjecture that the same is true for generic parameter deformations of the centralizers of the hyperoctahedral group and we reduce this conjecture to a conceptually much simpler numerical statement. Lastly, we address the trace classification problem for the Schur–Weyl dual of the halfliberated orthogonal group ${O}_{N}^{*}$, in which case extremal traces are always extensions of extremal traces on $ℂ\left[{S}_{\infty }×{S}_{\infty }\right]$. Our approach relies on methods developed by Vershik and Nikitin in [37].

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.218
Classification: 20G42,  05E10,  05A18
Keywords: Branching graph, minimal boundary, trace simplex, easy quantum groups.
Wahl, Jonas 1

1 Hausdorff Center for Mathematics Endenicher Allee 62 D-53115 Bonn, Germany
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Wahl, Jonas. Traces on diagram algebras II:  centralizer algebras of easy groups and new variations of the Young graph. Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 413-436. doi : 10.5802/alco.218. https://alco.centre-mersenne.org/articles/10.5802/alco.218/

[1] Aigner, Martin Catalan-like numbers and determinants, J. Combin. Theory Ser. A, Volume 87 (1999) no. 1, pp. 33-51 | DOI | MR | Zbl

[2] Aigner, Martin Enumeration via ballot numbers, Discrete Math., Volume 308 (2008) no. 12, pp. 2544-2563 | DOI | MR | Zbl

[3] Banica, Teodor; Speicher, Roland Liberation of orthogonal Lie groups, Adv. Math., Volume 222 (2009) no. 4, pp. 1461-1501 | DOI | MR | Zbl

[4] Banica, Teodor; Vergnioux, Roland Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier (Grenoble), Volume 60 (2010) no. 6, pp. 2137-2164 | DOI | Numdam | MR | Zbl

[5] Biane, Philippe Representations of symmetric groups and free probability, Adv. Math., Volume 138 (1998) no. 1, pp. 126-181 | DOI | MR | Zbl

[6] Bisch, Dietmar; Jones, Vaughan Algebras associated to intermediate subfactors, Invent. Math., Volume 128 (1997) no. 1, pp. 89-157 | DOI | MR | Zbl

[7] Borodin, Alexei; Olshanski, Grigori Representations of the infinite symmetric group, Cambridge Studies in Advanced Mathematics, 160, Cambridge University Press, Cambridge, 2016, vii+160 pages | DOI | MR | Zbl

[8] Brauer, Richard On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2), Volume 38 (1937) no. 4, pp. 857-872 | DOI | MR | Zbl

[9] Bufetov, Alexey; Gorin, Vadim Stochastic monotonicity in Young graph and Thoma theorem, Int. Math. Res. Not. IMRN (2015) no. 23, pp. 12920-12940 | DOI | MR | Zbl

[10] Colmenarejo, Laura; Orellana, Rosa; Saliola, Franco; Schilling, Anne; Zabrocki, Mike An insertion algorithm on multiset partitions with applications to diagram algebras, J. Algebra, Volume 557 (2020), pp. 97-128 | DOI | MR | Zbl

[11] Flake, Johannes; Maassen, Laura Semisimplicity and Indecomposable Objects in Interpolating Partition Categories, Int. Math. Res. Not. IMRN (2021), Paper no. rnab217 | DOI

[12] Freslon, Amaury; Weber, Moritz On the representation theory of partition (easy) quantum groups, J. Reine Angew. Math., Volume 720 (2016), pp. 155-197 | DOI | MR | Zbl

[13] Goodman, Frederick M.; de la Harpe, Pierre; Jones, Vaughan F. R. Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, 14, Springer Science & Business Media, 2012

[14] Gorin, Vadim The $q$-Gelfand–Tsetlin graph, Gibbs measures and $q$-Toeplitz matrices, Adv. Math., Volume 229 (2012) no. 1, pp. 201-266 | DOI | MR | Zbl

[15] Graham, John J.; Lehrer, Gustav I. Cellular algebras and diagram algebras in representation theory, Representation theory of algebraic groups and quantum groups (Adv. Stud. Pure Math.), Volume 40, Math. Soc. Japan, Tokyo, 2004, pp. 141-173 | DOI | MR | Zbl

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

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

[18] Jones, Vaughan F. R. Index for subfactors, Invent. Math., Volume 72 (1983) no. 1, pp. 1-25 | DOI | MR | Zbl

[19] Jones, Vaughan F. R. The Potts model and the symmetric group, Subfactors (Kyuzeso, 1993), World Sci. Publ., River Edge, NJ, 1994, pp. 259-267 | MR | Zbl

[20] Koike, Kazuhiko On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters, Adv. Math., Volume 74 (1989) no. 1, pp. 57-86 | DOI | MR | Zbl

[21] Lechner, Gandalf; Pennig, Ulrich; Wood, Simon Yang–Baxter representations of the infinite symmetric group, Adv. Math., Volume 355 (2019), p. 106769, 42 | DOI | MR | Zbl

[22] Martin, Paul P. The structure of the partition algebras, J. Algebra, Volume 183 (1996) no. 2, pp. 319-358 | DOI | MR | Zbl

[23] Martin, Paul P. The partition algebra and the Potts model transfer matrix spectrum in high dimensions, J. Phys. A, Volume 33 (2000) no. 19, pp. 3669-3695 | DOI | MR | Zbl

[24] 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

[25] Méliot, Pierre-Loïc Representation theory of symmetric groups, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2017, xvi+666 pages | DOI | MR | Zbl

[26] Nikitin, Pavel P. The centralizer algebra of the diagonal action of the group ${\mathrm{GL}}_{n}\left(ℂ\right)$ in a mixed tensor space, Journal of Mathematical Sciences, Volume 141 (2007) no. 4, pp. 1479-1493 | DOI

[27] Olshanski, Grigori Point processes related to the infinite symmetric group, The orbit method in geometry and physics (Marseille, 2000) (Progr. Math.), Volume 213, Birkhäuser Boston, Boston, MA, 2003, pp. 349-393 | DOI | MR | Zbl

[28] Orellana, Rosa C. On the algebraic decomposition of a centralizer algebra of the hyperoctahedral group, Algebraic structures and their representations (Contemp. Math.), Volume 376, Amer. Math. Soc., Providence, RI, 2005, pp. 345-357 | DOI | MR | Zbl

[29] Raum, Sven; Weber, Moritz The full classification of orthogonal easy quantum groups, Comm. Math. Phys., Volume 341 (2016) no. 3, pp. 751-779 | DOI | MR | Zbl

[30] Thoma, Elmar Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z., Volume 85 (1964), pp. 40-61 | DOI | MR | Zbl

[31] Timmermann, Thomas An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008, xx+407 pages | DOI | MR | Zbl

[32] Turaev, Vladimir G. Operator invariants of tangles, and R-matrices, Math. USSR, Izv., Volume 35 (1990) no. 2, pp. 411-444 | DOI | MR | Zbl

[33] Vershik, Anatoly M.; Kerov, Sergei V. Characters and factor representations of the infinite symmetric group, Dokl. Akad. Nauk SSSR, Volume 257 (1981) no. 5, pp. 1037-1040 | MR

[34] Vershik, Anatoly M.; Kerov, Sergei V. Characters and factor-representations of the infinite unitary group, Dokl. Akad. Nauk SSSR, Volume 267 (1982) no. 2, pp. 272-276 | MR | Zbl

[35] Vershik, Anatoly M.; Malyutin, Andrei V. Phase transition in the exit boundary problem for random walks on groups, Funct. Anal. Appl., Volume 49 (2015) no. 2, pp. 86-96 | DOI | MR | Zbl

[36] Vershik, Anatoly M.; Malyutin, Andrei V. The absolute of finitely generated groups: I. Commutative (semi)groups, Eur. J. Math., Volume 4 (2018) no. 4, pp. 1476-1490 | DOI | MR | Zbl

[37] Vershik, Anatoly M.; Nikitin, Pavel P. Traces on infinite-dimensional Brauer algebras, Funct. Anal. Appl., Volume 40 (2006) no. 3, pp. 165-172 | DOI | MR | Zbl

[38] Vershik, Anatoly M.; Nikitin, Pavel P. Description of the characters and factor representations of the infinite symmetric inverse semigroup., Funct. Anal. Appl., Volume 45 (2011) no. 1, pp. 13-24 | DOI | MR | Zbl

[39] Wahl, Jonas Traces on diagram algebras I: Free partition quantum groups, random lattice paths and random walks on trees, J. Lond. Math. Soc. (2), Volume 105 (2022) no. 4, pp. 2324-2372 | DOI | MR

[40] Wassermann, Antony J. Automorphic actions of compact groups on operator algebras, Ph. D. Thesis, University of Pennsylvania (1981)

[41] Weber, Moritz On the classification of easy quantum groups, Adv. Math., Volume 245 (2013), pp. 500-533 | DOI | MR | Zbl

[42] Wenzl, Hans On the structure of Brauer’s centralizer algebras, Ann. of Math. (2), Volume 128 (1988) no. 1, pp. 173-193 | DOI | MR | Zbl

[43] Weyl, Hermann The classical groups. Their invariants and representations, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997, xiv+320 pages (Fifteenth printing, Princeton Paperbacks) | MR

[44] Woronowicz, Stanisław L. Tannaka–Kreĭn duality for compact matrix pseudogroups. Twisted $\mathrm{SU}\left(N\right)$ groups, Invent. Math., Volume 93 (1988) no. 1, pp. 35-76 | DOI | MR | Zbl

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