In mathematics, a zonal polynomial is a multivariate

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...

. The zonal polynomials form a basis of the space of symmetric polynomials. They appear as zonal spherical functions of the Gelfand pairs

(S_,H_n)

(here,

H_n

is the hyperoctahedral group) and

(Gl_n(\mathbb),
O_n)

, which means that they describe canonical basis of the double class algebras

\mathbb_n \backslash S_ / H_n /math> and \mathbb_d(\mathbb)\backslash
M_d(\mathbb)/O_d(\mathbb) /math>.

They are applied in

multivariate statistics Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. Multivariate statistics concerns understanding the different aims and background of each of the dif ...

. The zonal polynomials are the

\alpha=2

case of the C normalization of the Jack function.

References

* Robb Muirhead, ''Aspects of Multivariate Statistical Theory'', John Wiley & Sons, Inc., New York, 1984. {{algebra-stub Homogeneous polynomials Symmetric functions