Abstract
The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a 0-Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the 0-Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restriction, and (anti-)involution twists of weak Bruhat interval modules.
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Notes
Following the notation in [13], \(\sigma \; {\bullet } \;\sigma ' = \sigma \cdot \sigma '[m]\) and \(\rho \; \overline{\bullet } \;\rho ' = \rho [n] \cdot \rho '\). And, there is a typo in [13, Theorem 3.8], where \(\beta = \beta ''[k]\cdot \beta '\) should appear as \(\beta = \beta '[n-k] \cdot \beta ''\).
In fact, in [14], the author considered the induction product and restriction of the simple and projective indecomposable modules over the 0-Hecke algebra of not only type A but also type B and D.
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Acknowledgements
The authors would like to thank Sarah Mason and Elizabeth Niese for helpful discussions on Remark 6. The authors also would like to thank Dominic Searles for helpful discussions on the 0-Hecke action on \(\varvec{R}^\sigma _\alpha \). The authors are grateful to the anonymous referee for careful readings of the manuscript and valuable advice
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All authors were supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF-2020R1F1A1A01071055). The second author was also supported by NRF Grant funded by the Korean Government (NRF-2019R1A2C4069647)
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Jung, WS., Kim, YH., Lee, SY. et al. Weak Bruhat interval modules of the 0-Hecke algebra. Math. Z. 301, 3755–3786 (2022). https://doi.org/10.1007/s00209-022-03025-4
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DOI: https://doi.org/10.1007/s00209-022-03025-4