Ribbon Schur polynomials

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#Ribbon Schur polynomials

The ribbon Schur polynomials are the skew Schur polynomials indexed by ribbons. In particular, Gessel’s fundamental expansion for skew Schur functions gives \[\schurS_\alpha(\xvec) = \sum_{T \in \SYT(\alpha)} \gessel_{D(T)}(\xvec),\] where $\alpha$ is a ribbon and $D(T)$ is the descent set of $T.$

There are $2^{n-1}$ different ribbons $\alpha$ with $n$ boxes, but the corresponding ribbon Schur functions $\schurS_\alpha$ are distinct, see [Rub10].

A. Riehl uses ribbon Schur identities to compute generating functions for permutations with prescribed descent sets [Rie08]. She also studies the dual basis to the ribbon Schur functions $Z_\lambda$ indexed by partitions, giving a combinatorial interpretation for the monomial coefficients of the dual basis and applications to expansions between arbitrary ribbon Schur functions and those indexed by partitions.

P. McNamara and S. v. Willigenburg study the Schur-positivity order on skew Schur functions, with particular attention to ribbons [MW09]. For ribbons with a fixed number of rows, they show that the corresponding subposet is convex. They also give a complete description for multiplicity-free ribbons: such a ribbon has at most two rows and at most two columns of length greater than one, and the resulting Schur-positivity poset is essentially a product of two chains.

#Complete homogeneous expansion

The ribbon Schur function $\schurS_\alpha$ can be expanded in the complete homogeneous basis as \[\sum_{\beta \geq \alpha} (-1)^{\length(\beta)-\length(\alpha)}\completeH_{\beta}\] where the sum is over coarsenings of $\alpha,$ see [Mou23]. A generalization to colored ribbon Schur functions can also be found in that reference.

#Products of ribbon Schur functions

For compositions $\alpha=(\alpha_1,\dotsc,\alpha_k)$ and $\beta=(\beta_1,\dotsc,\beta_\ell),$ let $\alpha\cdot\beta$ denote concatenation and let \[\alpha\odot\beta \coloneqq (\alpha_1,\dotsc,\alpha_{k-1},\alpha_k+\beta_1,\beta_2,\dotsc,\beta_\ell)\] denote near concatenation. Then ribbon Schur functions satisfy \[\schurS_\alpha\schurS_\beta = \schurS_{\alpha\cdot\beta}+\schurS_{\alpha\odot\beta},\] see [BTW06]. In that paper, these multiplicative relations are also used to describe the algebra generated by the ribbon Schur functions and to classify when two ribbon Schur functions are equal.

Example

The two ribbons with two boxes are the vertical domino $(1,1)$ and the horizontal domino $(2):$

${}$ $ {}$ ${} $ $ {}$

Their complete homogeneous expansions are \[\schurS_{(1,1)}=\completeH_1^2-\completeH_2 \qquad\text{and}\qquad \schurS_{(2)}=\completeH_2.\] The product rule gives, for instance, \[\schurS_{(1,1)}\schurS_{(2)} = \schurS_{(1,1,2)}+\schurS_{(1,3)}.\]

Bibliography

[BTW06]Louis J. Billera, Hugh Thomas and Stephanie Willigenburg. Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions. Advances in Mathematics, 204(1):204–240, 2006.

.bib

@article{BilleraThomasWilligenburg2006,
  author = {Billera, Louis J. and Thomas, Hugh and {van Willigenburg}, Stephanie},
  title = {Decomposable compositions, symmetric quasisymmetric functions and
    equality of ribbon {S}chur functions},
  year = {2006},
  journal = {Advances in Mathematics},
  volume = {204},
  number = {1},
  pages = {204--240},
  doi = {10.1016/j.aim.2005.05.014},
  url = {https://doi.org/10.1016/j.aim.2005.05.014},
  eprint = {math/0405434}
}

[MW09]Peter R. W. McNamara and Stephanie Willigenburg. Positivity results on ribbon Schur function differences. European Journal of Combinatorics, 30(5):1352–1369, 2009.

.bib

@article{McNamaraWilligenburg2009Ribbon,
  author = {McNamara, Peter R. W. and {van Willigenburg}, Stephanie},
  title = {Positivity results on ribbon {S}chur function differences},
  year = {2009},
  journal = {European Journal of Combinatorics},
  volume = {30},
  number = {5},
  pages = {1352--1369},
  doi = {10.1016/j.ejc.2008.09.026},
  url = {https://doi.org/10.1016/j.ejc.2008.09.026},
  eprint = {0712.2424}
}

[Mou23]Vassilis Dionyssis Moustakas. Descent representations and colored quasisymmetric functions. arXiv:2309.13615, 2023.

.bib

@article{Moustakas2023x,
Author = {Vassilis Dionyssis Moustakas},
Title = {Descent representations and colored quasisymmetric functions},
Year = {2023},
Eprint = {2309.13615},
  url = {https://arxiv.org/abs/2309.13615},
journal = {arXiv e-prints}
}

[Rie08]Amanda Riehl. Ribbon Schur functions and permutation patterns. University of California, San Diego, 2008.

.bib

@phdthesis{Riehl2008,
  author = {Amanda Riehl},
  title = {Ribbon {S}chur functions and permutation patterns},
  year = {2008},
  school = {University of California, San Diego},
  url = {https://escholarship.org/uc/item/5fv2k1v9}
}

[Rub10]Martin Rubey. The number of ribbon Schur functions. arXiv:1008.2501, 2010.

.bib

@article{Rubey2010x,
Author = {Martin Rubey},
Title = {The number of ribbon {S}chur functions},
Year = {2010},
Eprint = {1008.2501},
  url = {https://arxiv.org/abs/1008.2501},
journal = {arXiv e-prints}
}