Stałe strukturalne charakterów Jacka
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2018
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Structure constants of Jack characters
Abstract
W 1996 r. Goulden i Jackson wprowadzili rodzinę współczynników $( c_{\mu, \nu}^{\lambda} ) $ indeksowaną trójkami partycji, która pojawia się w rozwinięciu w szereg potęgowy pewnej sumy Cauchy'ego symetrycznych wielomianów Jacka $ J^{(\alpha )}_\pi $. Goulden i Jackson przypuszali, że za współczynnikami $c_{\mu, \nu}^{\lambda}$ ukryta jest kombinatoryka związana ze skojarzeniami. Postawiona przez nich hipoteza ,,O Skojarzeniach Jacka'' pozostaje do dzisiaj otwarta. Charaktery Jacka są uogólnieniem charakterów grup symetrycznych oraz obiektami dualnymi do wielomianów Jacka. W rozprawie doktorskiej badamy stałe strukturalne $ g_{\mu, \nu}^{\lambda} $ charakterów Jacka. Są one uogólnieniem stałych strukturalnych grup symetrycznych. Podajemy wzory na współczynniki najwyższych stopni wielomianów $ g_{\mu, \nu}^{\lambda} $ i $ c_{\mu, \nu}^{\lambda} $. Prezentujemy te rezultaty w kontekście hipotezy ,,O Skojarzeniach Jacka''. Adaptujemy probabilistyczne pojęcie kumulanty do struktury przestrzeni liniowej z dwoma mnożeniami. Prezentujemy formułę która wyraża pewien mieszany iloczyn jako sumę kumulant. Znalezione wyrażenie prowadzi do wniosków na temat stałych strukturalnych charakterów Jacka. Pokazujemy również, że nasza formuła może zostać uznana za odpowiednik formuły Leonova i Shiraeva.
In 1996 Goulden and Jackson introduced a family of coefficients $(c_{\mu, \nu}^{\lambda})$ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $J^{(\alpha )}_\pi$. Goulden and Jackson suggested that there is a combinatorics of matchings hidden behind the coefficients $c_{\mu,\nu}^{\lambda}$. This \emph{Matchings-Jack Conjecture} remains open. Jack characters are a generalization of the characters of the symmet\-ric groups, they provide a kind of dual information about the Jack polynomials. We investigate the structure constants $g_{\mu,\nu}^{\lambda}$ for Jack characters. They are a generalization of the connection coefficients for the symmetric groups. We give formulas for the top-degree part of $g_{\mu,\nu}^{\lambda}$ and $c_{\mu,\nu}^{\lambda}$. We present those results in context of Matchings-Jack Conjecture of Goulden and Jackson. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which expresses a given nested product with respect to those two multiplications as a sum of products of the cumulants. This formula leads to some conclusions about the structure constants of Jack characters. We also show that our formula may be understood as an analogue of Leonov--Shiraev's formula.
In 1996 Goulden and Jackson introduced a family of coefficients $(c_{\mu, \nu}^{\lambda})$ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $J^{(\alpha )}_\pi$. Goulden and Jackson suggested that there is a combinatorics of matchings hidden behind the coefficients $c_{\mu,\nu}^{\lambda}$. This \emph{Matchings-Jack Conjecture} remains open. Jack characters are a generalization of the characters of the symmet\-ric groups, they provide a kind of dual information about the Jack polynomials. We investigate the structure constants $g_{\mu,\nu}^{\lambda}$ for Jack characters. They are a generalization of the connection coefficients for the symmetric groups. We give formulas for the top-degree part of $g_{\mu,\nu}^{\lambda}$ and $c_{\mu,\nu}^{\lambda}$. We present those results in context of Matchings-Jack Conjecture of Goulden and Jackson. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which expresses a given nested product with respect to those two multiplications as a sum of products of the cumulants. This formula leads to some conclusions about the structure constants of Jack characters. We also show that our formula may be understood as an analogue of Leonov--Shiraev's formula.
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Wydział Matematyki i Informatyki
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Keywords
Jack, charaktery, kombinatoryka ennumeratywna, ennumerative combinatorics, mapy, maps, skojarzenia, matchings