Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants
Let $\lambda$ be a partition with no more than $n$ parts. Let $\beta$ be a weakly increasing $n$-tuple with entries from $\{ 1, ... , n \}$. The flagged Schur function in the variables $x_1, ... , x_n$ that is indexed by $\lambda$ and $\beta$ has been defined to be the sum of the content weight mono...
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| Format: | Article |
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Discrete Mathematics & Theoretical Computer Science
2021-04-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/6632/pdf |
| _version_ | 1827323730818236416 |
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| author | Robert A. Proctor Matthew J. Willis |
| author_facet | Robert A. Proctor Matthew J. Willis |
| author_sort | Robert A. Proctor |
| collection | DOAJ |
| description | Let $\lambda$ be a partition with no more than $n$ parts. Let $\beta$ be a
weakly increasing $n$-tuple with entries from $\{ 1, ... , n \}$. The flagged
Schur function in the variables $x_1, ... , x_n$ that is indexed by $\lambda$
and $\beta$ has been defined to be the sum of the content weight monomials for
the semistandard Young tableaux of shape $\lambda$ whose values are row-wise
bounded by the entries of $\beta$. Gessel and Viennot gave a determinant
expression for the flagged Schur function indexed by $\lambda$ and $\beta$;
this could be done since the pair $(\lambda, \beta)$ satisfied their
"nonpermutable" condition for the sequence of terminals of an $n$-tuple of
lattice paths that they used to model the tableaux. We generalize flagged Schur
functions by dropping the requirement that $\beta$ be weakly increasing. Then
for each $\lambda$ we give a condition on the entries of $\beta$ for the pair
$(\lambda, \beta)$ to be nonpermutable that is both necessary and sufficient.
When the parts of $\lambda$ are not distinct there will be multiple row bound
$n$-tuples $\beta$ that will produce the same set of tableaux. We accordingly
group the bounding $\beta$ into equivalence classes and identify the most
efficient $\beta$ in each class for the determinant computation. We recently
showed that many other sets of objects that are indexed by $n$ and $\lambda$
are enumerated by the number of these efficient $n$-tuples. We called these
counts "parabolic Catalan numbers". It is noted that the $GL(n)$ Demazure
characters (key polynomials) indexed by 312-avoiding permutations can also be
expressed with these determinants. |
| first_indexed | 2024-04-25T01:57:10Z |
| format | Article |
| id | doaj.art-26d900e0c0f8442db7cb997071656bad |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T01:57:10Z |
| publishDate | 2021-04-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-26d900e0c0f8442db7cb997071656bad2024-03-07T15:44:09ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502021-04-01vol. 23 no. 1Combinatorics10.46298/dmtcs.66326632Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinantsRobert A. ProctorMatthew J. WillisLet $\lambda$ be a partition with no more than $n$ parts. Let $\beta$ be a weakly increasing $n$-tuple with entries from $\{ 1, ... , n \}$. The flagged Schur function in the variables $x_1, ... , x_n$ that is indexed by $\lambda$ and $\beta$ has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape $\lambda$ whose values are row-wise bounded by the entries of $\beta$. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by $\lambda$ and $\beta$; this could be done since the pair $(\lambda, \beta)$ satisfied their "nonpermutable" condition for the sequence of terminals of an $n$-tuple of lattice paths that they used to model the tableaux. We generalize flagged Schur functions by dropping the requirement that $\beta$ be weakly increasing. Then for each $\lambda$ we give a condition on the entries of $\beta$ for the pair $(\lambda, \beta)$ to be nonpermutable that is both necessary and sufficient. When the parts of $\lambda$ are not distinct there will be multiple row bound $n$-tuples $\beta$ that will produce the same set of tableaux. We accordingly group the bounding $\beta$ into equivalence classes and identify the most efficient $\beta$ in each class for the determinant computation. We recently showed that many other sets of objects that are indexed by $n$ and $\lambda$ are enumerated by the number of these efficient $n$-tuples. We called these counts "parabolic Catalan numbers". It is noted that the $GL(n)$ Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants.https://dmtcs.episciences.org/6632/pdfmathematics - combinatorics05e05, 05a19 |
| spellingShingle | Robert A. Proctor Matthew J. Willis Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants Discrete Mathematics & Theoretical Computer Science mathematics - combinatorics 05e05, 05a19 |
| title | Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants |
| title_full | Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants |
| title_fullStr | Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants |
| title_full_unstemmed | Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants |
| title_short | Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants |
| title_sort | row bounds needed to justifiably express flagged schur functions with gessel viennot determinants |
| topic | mathematics - combinatorics 05e05, 05a19 |
| url | https://dmtcs.episciences.org/6632/pdf |
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