0 A ug 1 99 9 Positivity for special cases of ( q , t ) - Kostka coefficients and standard tableaux statistics
نویسنده
چکیده
We present two symmetric function operators H qt 3 and H qt 4 that have the property H qt 3 H (2 a 1 b) [X; q, t] = H (32 a 1 b) [X; q, t] and H qt 4 H (2 a 1 b) [X; q, t] = H (42 a 1 b) [X; q, t]. These operators are generalizations of the analogous operator H qt 2 and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, a µ (T) and b µ (T), on standard tableaux such that the q, t Kostka polynomials are given by the sum over standard tableaux of shape λ, K λµ (q, t) = T t aµ(T) q bµ(T) for the case when when µ is two columns or of the form (32 a 1 b) or (42 a 1 b). This provides proof of the positivity of the (q, t)-Kostka coefficients in the previously unknown cases of K λ(32 a 1 b) (q, t) and K λ(42 a 1 b) (q, t). The vertex operator formulas are used to give formulas for generating functions for classes of standard tableaux that generalize the case when µ is two columns.
منابع مشابه
Positivity for special cases of (q, t)-Kostka coefficients and standard tableaux statistics
We present two symmetric function operators H 3 and H qt 4 that have the property H mH(2a1b)(X; q, t) = H(m2a1b)(X; q, t). These operators are generalizations of the analogous operator H 2 and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, aμ(T ) and bμ(T ), on standard tableaux such that the q, t Kostka polynomials are given by the sum over stan...
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