The symmetric functions catalog

#Odd Schur functions

The odd Schur functions are introduced in [EKL12, EK12]. They constitute a basis for the space of odd symmetric functions.

For fixed \(n,\) the algebra of odd symmetric functions \(O\Lambda_n\) may be presented with generators \(\oddElementaryE_1,\dotsc,\oddElementaryE_n,\) where \(\oddElementaryE_0=1\) and \(\oddElementaryE_i=0\) for \(i\notin[0,n].\) The defining relations are \[\oddElementaryE_i\oddElementaryE_j = \oddElementaryE_j\oddElementaryE_i \quad\text{if } i+j \text{ is even},\] and \[\oddElementaryE_i\oddElementaryE_j +(-1)^i\oddElementaryE_j\oddElementaryE_i = \oddElementaryE_{j-1}\oddElementaryE_{i+1} +(-1)^i\oddElementaryE_{i+1}\oddElementaryE_{j-1} \quad\text{if } i+j \text{ is odd}.\] The stable algebra \(O\Lambda=\varinjlim O\Lambda_n\) is the \(q=-1\) specialization of the \(q\)-Hopf algebra construction of A. Ellis and M. Khovanov, and is a \(\setZ\)-graded Hopf superalgebra [EK12]. It contains odd analogues of the elementary, complete homogeneous, power-sum, monomial, and Schur bases.

There are several definitions of the odd Schur functions, \(\{\oddSchur_\lambda\},\) using divided difference operators or plactic relations. In [Ell12], it was shown that all the previous definitions coincide, and that we have the following tableau formula. \[\oddCompleteH_\mu = \sum_{T \in \SSYT(\lambda,\mu)} \sign(T_\lambda) \sign(T) \oddSchur_\lambda.\] Here, \(\sign(T)\) is the sign of the shortest permutation that sorts the reading word of \(T\) in an increasing fashion, and \(T_\lambda\) is the unique SSYT in \(\SSYT(\lambda,\lambda).\)

In [Ell12], a Littlewood–Richardson rule is proved for the odd Schur functions.

Bibliography

1. [Ell12]Alexander P. Ellis. The odd Littlewood–Richardson rule. Journal of Algebraic Combinatorics, 37(4):777–799, August 2012.
   .bib

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   @article{Ellis2012,
     doi = {10.1007/s10801-012-0389-6},
     url2 = {https://doi.org/10.1007/s10801-012-0389-6},
     year = {2012},
     month = aug,
     publisher = {Springer Science and Business Media {LLC}},
     volume = {37},
     number = {4},
     pages = {777--799},
     author = {Alexander P. Ellis},
     title = {The odd {L}ittlewood--{R}ichardson rule},
     journal = {Journal of Algebraic Combinatorics}
   }
   

2. [EK12]Alexander P. Ellis and Mikhail Khovanov. The Hopf algebra of odd symmetric functions. Advances in Mathematics, 231(2):965–999, October 2012.
   .bib

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   @article{EllisKhovanov2012,
     doi = {10.1016/j.aim.2012.04.031},
     url2 = {https://doi.org/10.1016/j.aim.2012.04.031},
     year = {2012},
     month = oct,
     publisher = {Elsevier {BV}},
     volume = {231},
     number = {2},
     pages = {965--999},
     author = {Alexander P. Ellis and Mikhail Khovanov},
     title = {The {H}opf algebra of odd symmetric functions},
     journal = {Advances in Mathematics}
   }
   

3. [EKL12]Alexander P. Ellis, Mikhail Khovanov and Aaron D. Lauda. The odd nilHecke algebra and its diagrammatics. International Mathematics Research Notices, 2014(4):991–1062, November 2012.
   .bib

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   @article{EllisKhovanovLauda2012,
     doi = {10.1093/imrn/rns240},
     url = {https://doi.org/10.1093/imrn/rns240},
     year = {2012},
     month = nov,
     publisher = {Oxford University Press ({OUP})},
     volume = {2014},
     number = {4},
     pages = {991--1062},
     author = {Alexander P. Ellis and Mikhail Khovanov and Aaron D. Lauda},
     title = {The Odd nil{H}ecke Algebra and its Diagrammatics},
     journal = {International Mathematics Research Notices}
   }