Hall–Littlewood polynomials

In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomials.

They were first defined indirectly by Philip Hall using the Hall algebra, and later defined directly by Dudley E. Littlewood (1961).

Definition

The Hall–Littlewood polynomial P is defined by

P

λ

(

x

1

,

…

,

x

n

;

t

)

=

(

∏

i

≥

0

∏

j

=

1

m

(

i

)

1

−

t

1

−

t

j

)

∑

w

∈

S

n

w

(

x

1

λ

1

⋯

x

n

λ

n

∏

i

<

j

x

i

−

t

x

j

x

i

−

x

j

)

,

{\displaystyle P_{\lambda }(x_{1},\ldots ,x_{n};t)=\left(\prod _{i\geq 0}\prod _{j=1}^{m(i)}{\frac {1-t}{1-t^{j}}}\right){\sum _{w\in S_{n}}w\left(x_{1}^{\lambda _{1}}\cdots x_{n}^{\lambda _{n}}\prod _{i<j}{\frac {x_{i}-tx_{j}}{x_{i}-x_{j}}}\right)},}

where λ is a partition of at most n with elements λi, and m(i) elements equal to i, and Sn is the symmetric group of order n!.

As an example,

P

42

(

x

1

,

x

2

;

t

)

=

x

1

4

x

2

2

+

x

1

2

x

2

4

+

(

1

−

t

)

x

1

3

x

2

3

{\displaystyle P_{42}(x_{1},x_{2};t)=x_{1}^{4}x_{2}^{2}+x_{1}^{2}x_{2}^{4}+(1-t)x_{1}^{3}x_{2}^{3}}

Specializations

We have that

P

λ

(

x

;

1

)

=

m

λ

(

x

)

{\displaystyle P_{\lambda }(x;1)=m_{\lambda }(x)}

,

P

λ

(

x

;

0

)

=

s

λ

(

x

)

{\displaystyle P_{\lambda }(x;0)=s_{\lambda }(x)}

and

P

λ

(

x

;

−

1

)

=

P

λ

(

x

)

{\displaystyle P_{\lambda }(x;-1)=P_{\lambda }(x)}

where the latter is the Schur P polynomials.

Properties

Expanding the Schur polynomials in terms of the Hall–Littlewood polynomials, one has

s

λ

(

x

)

=

∑

μ

K

λ

μ

(

t

)

P

μ

(

x

,

t

)

{\displaystyle s_{\lambda }(x)=\sum _{\mu }K_{\lambda \mu }(t)P_{\mu }(x,t)}

where

K

λ

μ

(

t

)

{\displaystyle K_{\lambda \mu }(t)}

are the Kostka–Foulkes polynomials.

Note that as

t

=

1

{\displaystyle t=1}

, these reduce to the ordinary Kostka coefficients.

A combinatorial description for the Kostka–Foulkes polynomials was given by Lascoux and Schützenberger,

K

λ

μ

(

t

)

=

∑

T

∈

S

S

Y

T

(

λ

,

μ

)

t

c

h

a

r

g

e

(

T

)

{\displaystyle K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}t^{\mathrm {charge} (T)}}

where "charge" is a certain combinatorial statistic on semistandard Young tableaux,

and the sum is taken over the set

S

S

Y

T

(

λ

,

μ

)

{\displaystyle SSYT(\lambda ,\mu )}

of all semi-standard Young tableaux T with shape λ and type μ.

See also

Hall polynomial

References

I.G. Macdonald (1979). Symmetric Functions and Hall Polynomials.