Geodesic
On the Schur positivity of \(\Delta_{e_{2}} e_n[X]\)
Qiu Dun1 ; Jeffrey B. Remmel1 ; Emily Sergel2 ; Guoce Xin3
1University of California San Diego
2University of Pennsylvania
3Capital Normal University
The electronic journal of combinatorics, Tome 25 (2018) no. 4
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Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson, and the second author have conjectured that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_k} e_n[X]$ is a polynomial in $\mathbb{N}[q,t]$. We present four proofs of a stronger statement in the case $k=2$; We show that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_2} e_n[X]$ has a positive expansion in terms of $q,t$-analogs.
DOI : 10.37236/7494
Classification : 05E05, 05E10
Mots-clés : Schur positivity, Macdonald polynomials, delta operator

Qiu Dun  1   ; Jeffrey B. Remmel  1   ; Emily Sergel  2   ; Guoce Xin  3

1 University of California San Diego
2 University of Pennsylvania
3 Capital Normal University 
@article{10_37236_7494,
     author = {Qiu Dun and Jeffrey B. Remmel and Emily Sergel and Guoce Xin},
     title = {On the {Schur} positivity of {\(\Delta_{e_{2}}} {e_n[X]\)}},
     journal = {The electronic journal of combinatorics},
     year = {2018},
     volume = {25},
     number = {4},
     doi = {10.37236/7494},
     zbl = {1398.05216},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/7494/}
}
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Qiu Dun; Jeffrey B. Remmel; Emily Sergel; Guoce Xin. On the Schur positivity of \(\Delta_{e_{2}} e_n[X]\). The electronic journal of combinatorics, Tome 25 (2018) no. 4. doi: 10.37236/7494

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