Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-24T09:48:16.168Z Has data issue: false hasContentIssue false
  • English
  • Français

DIAGRAM ALGEBRAS, DOMINANCE TRIANGULARITY AND SKEW CELL MODULES

Part of: Algebraic combinatorics Linear algebraic groups and related topics

Published online by Cambridge University Press:  17 October 2017

CHRISTOPHER BOWMAN ,
JOHN ENYANG  and
FREDERICK GOODMAN Open the ORCID record for FREDERICK GOODMAN [Opens in a new window]
CHRISTOPHER BOWMAN
Affiliation:
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7FS, UK email C.D.Bowman@kent.ac.uk
JOHN ENYANG
Affiliation:
Department of Mathematics, City University London, London, UK email john.enyang.1@city.ac.uk
FREDERICK GOODMAN*
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA email frederick-goodman@uiowa.edu
*
✉frederick-goodman@uiowa.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present an abstract framework for the axiomatic study of diagram algebras. Algebras that fit this framework possess analogues of both the Murphy and seminormal bases of the Hecke algebras of the symmetric groups. We show that the transition matrix between these bases is dominance unitriangular. We construct analogues of the skew Specht modules in this setting. This allows us to propose a natural tableaux theoretic framework in which to study the infamous Kronecker problem.

Keywords

cellular algebrasdiagram algebras

MSC classification

Primary: 20G05: Representation theory
Secondary: 05E10: Combinatorial aspects of representation theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 1 , February 2018 , pp. 13 - 36
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

We would like to thank the Royal Commission for the Exhibition of 1851 and EPSRC grant EP/L01078X/1 for financial support.

References

Bowman, C., De Visscher, M. and Enyang, J., ‘Simple modules for the partition algebra and monotone convergence of Kronecker coefficients’, Int. Math. Res. Notices, to appear,https://doi.org/10.1093/imrn/rnx095.Google Scholar
Bowman, C., De Visscher, M. and Orellana, R., ‘The partition algebra and the Kronecker coefficients’, Trans. Amer. Math. Soc. 367(5) (2015), 3647–3667.CrossRefGoogle Scholar
Bowman, C., Enyang, J. and Goodman, F., ‘The cellular second fundamental theorem of invariant theory for classical groups’, Preprint 2016, arXiv:1610.0900.Google Scholar
Enyang, J. and Goodman, F. M., ‘Cellular bases for algebras with a Jones basic construction’, Algebr. Represent. Theory 20(1) (2017), 71–121.Google Scholar
Geetha, T. and Goodman, F. M., ‘Cellularity of wreath product algebras and A-Brauer algebras’, J. Algebra 389 (2013), 151–190.CrossRefGoogle Scholar
Goodman, F. M. and Graber, J., ‘Cellularity and the Jones basic construction’, Adv. Appl. Math. 46(1–4) (2011), 312–362.Google Scholar
Goodman, F. M. and Graber, J., ‘On cellular algebras with Jucys Murphy elements’, J. Algebra 330 (2011), 147–176.CrossRefGoogle Scholar
Goodman, F. M., Kilgore, R. and Teff, N., ‘A cell filtration of the restriction of a cell module’, Preprint, 2015, arXiv:1504.02136.Google Scholar
Graham, J. J. and Lehrer, G. I., ‘Cellular algebras’, Invent. Math. 123(1) (1996), 1–34.Google Scholar
James, G. D., The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, 682 (Springer, Berlin, 1978).CrossRefGoogle Scholar
Mathas, A., Iwahori–Hecke Algebras and Schur Algebras of the Symmetric Group, University Lecture Series, 15 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
Mathas, A., ‘Seminormal forms and Gram determinants for cellular algebras’, J. reine angew. Math. 619 (2008), 141–173. With an appendix by Marcos Soriano.Google Scholar
Mathas, A., ‘Restricting Specht modules of cyclotomic Hecke algebras’, Sci. China Math. (2017). https://doi.org/10.1007/s11425-016-9032-2.CrossRefGoogle Scholar
Murphy, G. E., ‘On the representation theory of the symmetric groups and associated Hecke algebras’, J. Algebra 152(2) (1992), 492–513.Google Scholar
Murphy, G. E., ‘The representations of Hecke algebras of type A n ’, J. Algebra 173(1) (1995), 97–121.Google Scholar
Okounkov, A. and Vershik, A., ‘A new approach to representation theory of symmetric groups’, Selecta Math. (N.S.) 2(4) (1996), 581–605.Google Scholar
Okounkov, A. and Vershik, A., ‘A new approach to the representation theory of the symmetric groups. II’, J. Math. Sci. 131 (2005), 5471–5494.Google Scholar
Pak, I. and Panova, G., ‘Bounds on certain classes of Kronecker and q-binomial coefficients’, J. Combin. Theory Ser. A 147 (2017), 1–17.Google Scholar
Peel, M. H. and James, G. D., ‘Specht series for skew representations of symmetric groups’, J. Algebra 56(2) (1979), 343–364.Google Scholar
Stanley, R. P., Enumerative Combinatorics,  Vol. 2, Cambridge Studies in Advanced Mathematics, 62 (Cambridge University Press, Cambridge, 1999). With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.Google Scholar