Date of Completion
5-15-2025
Degree Type
Honors Thesis
Discipline
Mathematics (MATH)
First Advisor
Joshua Hallam
Abstract
We introduce a family of descent-preserving, sign-reversing involutions on encodings of standard immaculate tableaux which, in the two-row case, identifies pairs standard immaculate tableaux, leaving exactly those satisfying the conditions of standard Young composition tableaux fixed. Using this result, we derive the first explicit signed decomposition of the Young quasisymmetric Schur functions into dual immaculate quasisymmetric functions, confirming the conjectured decomposition, by Allen-Hallam-Mason, in the two-row setting. Along the way, considering the number of descent elements, we obtain counted refinements of standard immaculate tableaux and standard Young composition tableaux with shapes (n,n+1) and (n+1,n), corresponding to the Catalan and Narayana numbers. Finally, in an attempt to prove the case with more rows, we generalize word encodings of standard immaculate tableaux and standard young composition tableaux and we construct a group which is conjectured to demonstrate the decomposition in the three-row case.
Recommended Citation
Butts, Gavin and Hallam, Joshua, "Towards the Signed Decomposition of Young Quasisymmetric Schur Functions into Dual Immaculate Quasisymmetric Functions" (2025). Honors Thesis. 595.
https://digitalcommons.lmu.edu/honors-thesis/595
