UDC 517.514+517.562.+517.564

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR L. A. LYUSTERNIK

ON SOME SYSTEMS OF SYMMETRIC FUNCTIONS

We retain the notation of (1): \(x_i\) are Cartesian coordinates (c.c.) in \(E_n\); \(e_j\), \(j=0,1,\ldots,n\), are unit vectors in \(E_n\), the vertices of the regular \(n\)-dimensional simplex \(D_n\); \(y_j\), \(j=0,1,\ldots,n\), are the “barycentric” coordinates (b.c.) of the vector \(Y=[y_j]=\sum_j y_j e_j\) in \(E_n\); \([y_j]=[y_j+y]\), \(y\) arbitrary;

\[ y_{\mathrm{cp}}=\frac{1}{n+1}\sum y_j,\qquad y'_j=y_j-y_{\mathrm{cp}} \]

are the canonical b.c. of the vector \([y_j]=[y'_j]\); \(\Omega_n\) is the lattice in \(E_n\), the set of vectors \(K=[k_j]\) with integral b.c. \(k_j\); \(W_n\) is the Dirichlet domain of \(\Omega_n\), its volume \(|W_n|=(n+1)^{(n-1)/2}n^{-n/2}\);

\[ \sum_j=\sum_{j=0}^n;\qquad \prod_j=\prod_{j=0}^{\infty};\qquad \sum_m=\sum_{m=-\infty}^{\infty};\qquad \frac{1}{m!}=0\quad \text{for } m<0. \]

Let \(K=[k_j]\subset \Omega_n\)

\[ U_K^{(n)}(x)=U_{(k_j)}^{(n)}(x)=U_{(k_0,k_1,\ldots,k_n)}^{(n)}(x) =\sum_m \prod_j \frac{x^{m+k_j}}{(m+k_j)!}. \tag{1} \]

For \(Z=[z_j]=[z'_j]\), \(Y=[y_j]=[y'_j]\),

\[ \langle Z,Y\rangle=\sum_j z_j y'_j=\sum_j y_j z'_j . \]

1. Let there be given \(m\) indices \(i\) (for example, \(i=0,1,\ldots,m-1\) or \(i=1,2,\ldots,m\)) and \(m\) numbers \(a_i\) with indices \(i\); let \(s\) be a permutation of these indices, \(i\to si\). Then \(sa_i=a_{si}\); \(S_m\) is the group of the \(m!\) permutations \(s\). If, among the numbers \(a_i\), \(l_p\) are equal to \(p\), \(\sum_p l_p=m\), then \(a(a_i)=\prod_p l_p!\) is the number of permutations \(s\subset S_m\) for which all \(sa_i=a_i\). We shall, in particular, consider the group \(S_{n+1}\) of permutations \(s\) of the \((n+1)\) indices \(j=0,1,\ldots,n\) of the b.c. of the vector \(Y=[y_j]\); we denote \(sY=[sy_j]=[y_{sj}]\); thus \(s\) may be regarded as a transformation of \(E_n\): \(Y\to sY\), and the group \(S_{n+1}\) as the group of transformations of the “simplex \(D_n\)” of the space \(E_n\). Each vector \(K=[k_j]\subset\Omega_n\) passes under transformations \(s\subset S_{n+1}\) into vectors \(sK\), differing only in the order of the b.c.; the number of distinct ones among them is \((n+1)!/a(k_j)\); we denote the set of these vectors by \(\mathcal K=\{k_j\}\); \(a(\mathcal K)=a(k_j)\), and the set of all \(\mathcal K\)’s by \(\widetilde{\Omega}_n\).

2. The functions \(U_K^{(n)}(x)=U_{(k_1\ldots k_n)}(x)\) are symmetric in the \(k_j\). We shall denote by \(U_{\mathcal K}^{(n)}(x)\) all functions \(U_K^{(n)}(x)\) that are equal to one another, \(K\subset\mathcal K\).

Let \(\prod_j t_j=1\), \(K=[k_j]\), \(t^K=\prod_j t_j^{k_j}\); for \(K\subset\mathcal K=\{k_j\}\),

\[ A_n(\mathcal K,t)=A_n(t_j,k_j)=\sum_{s\subset S_{n+1}} t^{sK}. \tag{2} \]

The generating formula for \(U_K^{(n)}(x)\) from (1) takes the form

\[ \exp\left(x\sum_j t_j\right) = \sum_{\mathcal K\subset\widetilde{\Omega}_n} \frac{1}{a(\mathcal K)}A_n(\mathcal K,t)\,U_{\mathcal K}^{(n)}(x). \tag{3} \]

For \(t_j=e^{y'_j}\left(\sum_j y'_j=0,\,[y'_j]=Y'\right)\)

\[ B_n(\mathcal K,Y')=B_n(k_j,y'_j)=A_n(k_j,t_j)=\sum_{s\subset S_{n+1}} e^{\langle sK,Y\rangle}, \tag{4} \]

\[ C_n(\mathcal K,Y)=B_n(\mathcal K,2\pi iY). \tag{5} \]

Replacing \(t_j\) in (3) by \(e^{y'_j}\) \((e^{2\pi i y_j})\) leads to replacing \(A_n(\mathcal K,t)\) by \(B_n(\mathcal K,Y)\) \((C_n(\mathcal K,Y))\). The integral representation from (1) for \(U_{\mathcal K}^{(n)}(x)\) takes the form

\[ U_{\mathcal K}^{(n)}(x)=\frac{1}{|d_n|}\int_{d_n}\cdots\int \exp\left(\sum_j xe^{2\pi i y_j}\right)\overline{C_n(\mathcal K,Y)}\,d\omega_n; \tag{6} \]

\(d_n\) is the set of points \(W_n\) for which \(y_0\leq y_1\leq\cdots\leq y_n\); \(|d_n|=\dfrac{1}{(n+1)!}|W_n|\). Note that the system \(C_n(\mathcal K,Y)\), \(\mathcal K\subset\Omega_n\), as functions of \(Y\), forms a complete orthogonal system on \(d_n\), and every function \(f(Y)\) with integrable square on \(d_n\) is expanded there in the Fourier series

\[ f(Y)=\sum_{\mathcal K\subset\Omega_n}\frac{C_{\mathcal K}}{a(\mathcal K)}C_n(\mathcal K,Y),\qquad C_{\mathcal K}=\frac{1}{|d_n|}\int_{d_n}\cdots\int f(Y)\overline{C_n(\mathcal K,Y)}\,d\omega_n. \tag{7} \]

Remark. Every function \(f(Y)\) defined on \(d_n\) is extended by the equalities \(f(sY)=f(Y)\), \(s\subset S_{n+1}\), \(f(Y+e_j)=f(Y)\) to a function symmetric with respect to the b.c. in \(E_n\) with vector periods \(e_j\) (such are \(C_n(\mathcal K,Y)\)).

1. We consider o.f.—homogeneous forms and s.f.—symmetric o.f. in \(E_n\); when passing from d.c. to b.c. and conversely, the homogeneity and degree of an o.f. do not change. We distinguish s.f.d. and s.f.b.—s.f. with respect to d.c. and b.c. in \(E_n\). Sph.f.—spherical functions—are o.f. satisfying Laplace’s equation (simultaneously with respect to d.c. and b.c.); s.sph.d. (and s.sph.b.) are simultaneously sph.f. and s.f.d. (s.f.b.). \([\text{o.f.}_{p,n}]\), \([\text{s.f.b.}_{p,n}]\), etc. are linear systems of all o.f. (s.f.b., etc.) in \(E_n\) of degree \(p\); their dimensions are \(\dim[\text{o.f.}_{p,n}]\), etc. In what follows \(p,n,m,l,p_i\) are natural numbers.

We consider representations of \(p\) by sums

\[ p=\sum_{i=1}^{l}p_i \]

under restrictions on \(p_i\) and \(l\): a) \(p_i\geq m\); b) \(p_i\leq n\); c) \(p_i\neq k\); d) \(l\leq n\).

We denote the number of such representations of \(p\) under the conditions: a) by \(\alpha(p,m)\); a) and b) by \(\alpha(p,m,n)\); a), b), and c) by \(\alpha(p,m,n,k)\); a) and d) by \(\beta(p,m,n)\).

Obviously, for \(n\geq p\),
\(\alpha(p,m)=\alpha(p,m,n)=\beta(p,m,n)\). Further,
\(\beta(p,1,n)=\alpha(p,1,n)\), \(\beta(p,m,n)>\alpha(p,m,n)\) for \(m>1,\ n<p\);
\(\alpha(p,m,n,k)=\alpha(p,m,n)-\alpha(p-k,m,n)\).

As \(p\to\infty\), \(m<n\),

\[ \alpha(p,m,n)=\frac{(m-1)!}{n!(n-m)!}\,p^{\,n-m}+O(p^{\,n-m-1}). \]

\[ \dim[\text{s.f.d.}_{p,n}]=\alpha(p,1,n);\qquad \dim[\text{s.f.b.}_{p,n}]=\alpha(p,2,n+1); \]

\[ \dim[\text{s.sph.d.}_{p,n}]=\alpha(p,1,n,2);\qquad \dim[\text{s.sph.b.}_{p,n}]=\alpha(p,3,n+1), \]

\[ \lim_{p\to\infty}\frac{\dim[\text{s.f.d.}_{p,n}]}{\dim[\text{o.f.}_{p,n}]} = \lim_{p\to\infty}\frac{\dim[\text{s.sph.d.}_{p,n}]}{\dim[\text{sph.f.}_{p,n}]} =\frac{1}{n!}, \]

\[ \lim_{p\to\infty}\frac{\dim[\text{s.f.b.}_{p,n}]}{\dim[\text{o.f.}_{p,n}]} = \lim_{p\to\infty}\frac{\dim[\text{s.sph.b.}_{p,n}]}{\dim[\text{sph.f.}_{p,n}]} =\frac{1}{(n+1)!}. \]

All symmetric functions of \(m\) variables \(z_1, z_2, \ldots, z_m\) of degree \(p\) are linear combinations of the simplest such symmetric functions \(T^{(m)}_{(r_i)}=T^{(m)}_{r_1,r_2,\ldots,r_m}\), “generated” by the products \(\prod_i z_i^{r_i}\):

\[ T^{(m)}_{(r_i)}(z_i)=\frac{1}{a(r_i)}\sum_{s\subset S_m}\prod_i (s z_i)^{r_i},\qquad \sum_i r_i=p,\qquad r_i\geqslant 0. \]

In \([\text{s.f. } d_{n,p}]\) the forms \(T^{(n)}_{(r_i)}\) in the variables \(x_i\), \(\sum_i r_i=p\), form a basis.

In \([\text{s.f. } b_{n,p}]\) all forms are expressed linearly in terms of \(T^{(n+1)}_{(r_j)}(y'_j)\), \(\sum_j r_j=p\), where \(r_j\ne 1\); for \(n+1\geqslant p\) they form a basis, while for \(n+1<p\) there exist \(\beta(p,2,n+1)-\alpha(p,2,n+1)>0\) linear relations among them. For example, in \([\text{s.f. } b_{n,5}]\) for \(n\geqslant 4\) the basis is formed by
\(T^{(n+1)}_5(y'_j)=\sum_j y_j'^5\) and
\(T^{(n+1)}_{3,2}=\sum_{i\ne j} y_i'^3 y_j'^2\)
(we omit the indices \(r_j=0\) in the notation \(T^{(n+1)}_{(r_j)}\)). If \(n<4\), \((T^{(n+1)}_5-5T^{(n+1)}_{3,2})(y'_j)=0\); for \(n=1\), \(T^{(n+1)}_5=T^{(n+1)}_3=0\).

The symmetrized exponential functions \(B_n(\mathcal K,Y)=B_n(k_j,y'_j)\) are expanded in a series in the s.f.b. \(\bigl([T]^2=T(k'_j)T(y'_j)\bigr)\)

\[ B^{(n)}(\mathcal K,Y)=(n+1)!+\sum_{p=2}^{\infty} a_{p,n}(k_j,y'_j), \]

\[ a_{p,n}(k_j,y'_j)= \sum_{r_j\geqslant 0}\cdots \sum_{\sum_j r_j=p} \frac{a(r_j)\,p!}{\prod_j r_j!}\,[T^{(n+1)}_{(r_j)}]^2 . \tag{8} \]

Expressing all \(T^{(n+1)}_{(k_j)}(y'_j)\) in terms of the basis functions, we obtain for \(p=2,3,4,5\) (when \(n+1<p\) the terms containing the factor \((n-k)!\), where \(k>n\), vanish):

\[ a_{2,n}(k_j,y'_j)=(n+1)(n-1)![T^{(n+1)}_2]^2, \]

\[ a_{3,n}(k_j,y'_j)=(n+1)^2(n-2)![T^{(n+1)}_3]^2, \]

\[ \begin{aligned} a_{4,n}(k_j,y'_j) &=(n+4)(n-1)![T^{(n+1)}_4]^2 +12(n-1)![T^{(n+1)}_{2,2}]^2 \\ &\quad +(6n-3)(n-3)![T^{(n+1)}_4-2T^{(n+1)}_{2,2}]^2, \end{aligned} \]

\[ \begin{aligned} a_{5,n}(k_j,y'_j) &=(n-2)!\{(n+5)(n-1)[T^{(n+1)}_5]^2 +10(n+5)[T^{(n+1)}_{3,2}]^2 \\ &\quad +10[T^{(n+1)}_5-T^{(n+1)}_{3,2}]^2\} +(10n-14)(n-4)![T^{(n+1)}_5-5T^{(n+1)}_{3,2}]^2 . \end{aligned} \]

Moscow State University
named after M. V. Lomonosov

Received
12 X 1967

CITED LITERATURE

1. L. A. Lyusternik, DAN, 177, No. 5 (1967).