The symmetric functions catalog

#Zonal symmetric functions

The zonal symmetric functions were introduced by L.-K. Hua in [Hua63]. For an introduction, see [Chapter 7, Mac95].

The zonal symmetric functions \(\zonal_\lambda(\xvec)\) associated with the Gelfand pair \((\symS_{2n},H_n)\) are given by the specialization \(\alpha=2\) in the Jack symmetric functions. That is, \(\zonal_\lambda(\xvec) = \jackJ_\lambda(\xvec;2).\) L. Jiu and C. Koutschan give algorithms and implementation details for calculating zonal polynomials [JK20].

Here, \(H_n\) is the hyperoctahedral group of degree \(n,\) given by the centralizer of the simple transpositions \((12),(34),\dotsc,(2n-1,2n).\) Its order is \(|H_n| = 2^n n!.\)

#Power sum expansion

In [FS11], the authors present a formula for \(\zonal_\lambda(\xvec)\) as a signed sum over \(T\)-admissible pair-partitions. \[\zonal_\lambda(\xvec) = \sum_{(S_1,S_2) \text{ T-admissible}} (-1)^{L(S_1,S_2)} \powerSum_{L(S_1,S_2)}(\xvec).\] Here, \(T\) is the standard Young tableau of shape \(2\lambda,\) with \(1,2,\dotsc,2\lambda_1\) in the first row, and so on. The expression \(L(S_1,S_2)\) is the sizes of the components of certain bipartite graphs, where every vertex has degree \(2,\) there are \(2n\) edges labeled \(1,\dotsc,2n\) and \(\{i,j\} \in S_c\) if and only if edges \(i\) and \(j\) share a vertex with color \(c \in \{1,2\}.\)

Pair-partitions can be interpreted as a disjoint product of transpositions, and thus as elements in \(\symS_{2n}.\) Let \(S = \{\{1,2\},\{3,4\},\dotsc,\{2n-1,2n\}\}.\) Let \(T\) be a standard Young tableau. The pair \((S_1,S_2)\) is \(T\)-admissible if \(S \circ S_1\) preserves the rows of \(T,\) and \(S_2\) preserves the columns.

Bibliography

1. [FS11]Valentin Féray and Piotr Śniady. Zonal polynomials via Stanley’s coordinates and free cumulants. Journal of Algebra, 334(1):338–373, May 2011.
   .bib

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   @article{FeraySniady2011,
     doi = {10.1016/j.jalgebra.2011.03.008},
     year = {2011},
     month = may,
     publisher = {Elsevier {BV}},
     volume = {334},
     number = {1},
     pages = {338--373},
     author = {Valentin F{\'{e}}ray and Piotr {\'{S}}niady},
     title = {Zonal polynomials via {S}tanley's coordinates and free cumulants},
     journal = {Journal of Algebra}
   }
   

2. [Hua63]L. K. Hua. Harmonic analysis of functions of several complex variables in the classical domains (translations of mathematical monographs). American Mathematical Society, 1963.
   .bib

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   @book{Hua1963,
     Author = {L. K. Hua},
     Title = {Harmonic Analysis of Functions of Several Complex Variables in the 
     Classical Domains (Translations of Mathematical Monographs)},
     Publisher = {American Mathematical Society},
     Year = {1963},
     ISBN = {0821815563}
   }
   

3. [JK20]Lin Jiu and Christoph Koutschan. Calculation and properties of zonal polynomials. Mathematics in Computer Science, 14(3):623–640, 2020.
   .bib

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   @article{JiuKoutschan2020,
     author = {Lin Jiu and Christoph Koutschan},
     title = {Calculation and properties of zonal polynomials},
     year = {2020},
     journal = {Mathematics in Computer Science},
     volume = {14},
     number = {3},
     pages = {623--640},
     doi = {10.1007/s11786-020-00458-0},
     eprint = {2001.11599}
   }
   

4. [Mac95]Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford mathematical monographs. The Clarendon Press, Oxford University Press, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
   .bib

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   @book{Macdonald1995,
   	address = {New York},
   	author = {Ian G. Macdonald},
   	edition = {Second},
   	isbn = {0-19-853489-2},
   	mrclass = {05E05 (05-02 20C30 20C33 20K01 33C80 33D80)},
   	note = {With contributions by A. Zelevinsky, Oxford Science Publications},
   	pages = {x+475},
   	publisher = {The Clarendon Press, Oxford University Press},
   	series = {Oxford Mathematical Monographs},
   	title = {Symmetric functions and {H}all polynomials},
   	year = {1995}
   }