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We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions. We construct a cyclic, indecomposable 0-Hecke algebra module for these functions. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution on the ring of quasisymmetric functions. We give an explicit description of the effect of on the associated 0-Hecke modules, via the poset induced by the 0-Hecke action on standard immaculate tableaux. This remarkable poset reveals other 0-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf-Searles (2019). Like the dual immaculate function, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, corresponding to a specific descent set. We give a complete combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets, and showing thatallthe possible variations for generating functions of tableaux occur as characteristics of the 0-Hecke modules determined by these descent sets.
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P -Partitions and Quasisymmetric Power Sums
Abstract The $$(P, \omega )$$-partition generating function of a labeled poset $$(P, \omega )$$ is a quasisymmetric function enumerating certain order-preserving maps from $$P$$ to $${\mathbb{Z}}^+$$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $$\{\psi _{\alpha }\}$$. Using this expansion, we show that connected, naturally labeled posets have irreducible $$P$$-partition generating functions. We also show that series-parallel posets are uniquely determined by their partition generating functions. We conclude by giving a combinatorial interpretation for the coefficients of the $$\psi _{\alpha }$$-expansion of the $$(P, \omega )$$-partition generating function akin to the Murnaghan–Nakayama rule.
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- Award ID(s):
- 1700302
- PAR ID:
- 10277611
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnz375
