Reports of the Academy of Sciences of the USSR

1. Volume 160, No. 3

MATHEMATICS

DOAN KUINH

POINCARÉ POLYNOMIALS OF SOME COMPACT HOMOGENEOUS SPACES

(Presented by Academician A. I. Mal'cev, 7 VII 1964)

1°. Let \(\mathcal G\) be a connected compact Lie group; \(\mathcal T\) its maximal torus; \(G, T\) the corresponding Lie algebras; \(I_{S(G)}\) the algebra of all polynomials on \(T\) invariant with respect to the Weyl group. Then \(I_{S(G)}\) is a free algebra with \(r\) (\(r\) is the rank of \(\mathcal G\)) algebraically independent homogeneous generators \(P_{k_1}, P_{k_2}, \ldots, P_{k_r}\) (\(k_i\) is the degree of \(P_{k_i}\)), and the Poincaré polynomial of the group \(\mathcal G\) has the form

\[ P(\mathcal G,t)=(1+t^{2k_1-1})(1+t^{2k_2-1})\cdots(1+t^{2k_r-1}). \]

Theorem. Let \(G\) be a compact exceptional simple Lie algebra; let \(p_0(G)\) be its linear representation of least dimension; denote the weights of the representation \(p_0(G)\) by \(\Lambda_1\Lambda_2,\ldots,\Lambda_n\) (\(n\) is the dimension of \(p_0\)).

Then

\[ P_{k_i}=\sum_{\alpha=1}^{n}\Lambda_\alpha^{k_i} \]

for the following \(k_i\) form generating polynomials of the algebra \(I_{S(G)}\) (in parentheses are the values of \(k_i\)):

\[ G_2(2,6);\quad F_4(2,6,8,12);\quad E_6(2,5,6,8,9,12); \]

\[ E_7(2,6,8,10,12,14,18);\quad E_8(2,8,12,14,18,20,24,30). \]

(These \(P_{k_i}\) coincide with the polynomials considered by Coxeter \((^1)\) for \(E_6\) and by Takeuchi \((^2)\) for \(F_4\).)

We note that an analogue of this theorem, as is known, is also valid for \(A_r, B_r, C_r\). For \(D_r\), besides

\[ P_{k_i}=\sum_{\alpha=1}^{2r}\Lambda_\alpha^{k_i}, \]

one must also take \(P'_r=\Lambda_1\Lambda_2\cdots\Lambda_r\).

2°. Let a homogeneous space \(\mathcal G/\mathcal U\) be given, where \(\mathcal G\) is a connected compact Lie group, \(\mathcal U\) its connected closed subgroup; \(U\subset G\) are the corresponding Lie algebras. Denote the restriction of \(I_{S(G)}\) to \(U\) by \(p:I_{S(G)}\to I_{S(U)}\). If the rank of \(\mathcal G=R\), the rank of \(U=r\), and all \(p(P_{k_\alpha})\) \((r+1\leq \alpha\leq R)\) belong to the ideal in \(I_{S(U)}\) generated by \(p(P_{k_i})\) \((1\leq i\leq r)\) (where \(P_{k_1},P_{k_2},\ldots,P_{k_R}\) are generating polynomials of the algebra \(I_{S(G)}\)), then we shall say that \(\mathcal U\) is in normal position in \(\mathcal G\). In this case the Poincaré polynomial of the space \(\mathcal G/\mathcal U\) has the form

\[ P(\mathcal G/\mathcal U,t)= \frac{(1-t^{2k_1})(1-t^{2k_2})\cdots(1-t^{2k_r})} {(1-t^{2l_1})(1-t^{2l_2})\cdots(1-t^{2l_r})} \times \]

\[ \times(1+t^{2k_{r+1}-1})(1+t^{2k_{r+2}-1})\cdots(1+t^{2k_R-1}), \]

where \(l_1,l_2,\ldots,l_r\) are the degrees of the generating polynomials of the algebra \(I_{S(U)}\).

The case \(R=r\) (then we have Hirzebruch’s formula) or the case \(p:I_{S(G)}\to I_{S(U)}\) is an epimorphism (then \(\mathcal U\) is completely nonhomologous to 0 in \(\mathcal G\) and \(P(\mathcal G/\mathcal U,t)=\dfrac{P(\mathcal G,t)}{P(\mathcal U,t)}\)) is a special case of normal position.

\(3^\circ\). Consider compact homogeneous nonsymmetric spaces \(\mathfrak G/\mathfrak U\) with an irreducible rotation group. All these spaces were found in the papers of O. V. Manturov \((^{3a,b})\).

For these spaces \(\mathfrak U\) is in normal position in \(\mathfrak G\), except for the following 4 cases (in the diagrams the embedding \(U \to G\) is indicated):

a)
\[ \overset{1}{\circ}-\circ-\cdots-\circ \times \overset{1}{\circ}-\circ-\cdots-\circ \to SU\bigl((n_1+1)(n_2+1)\bigr) \]
\[ \underbrace{\hspace{3.0cm}}_{n_1}\qquad \underbrace{\hspace{3.0cm}}_{n_2} \]

or

\[ \overset{1}{\circ}-\circ-\cdots-\circ \times \circ-\circ-\cdots-\overset{1}{\circ} \to SU\bigl((n_1+1)(n_2+1)\bigr) \]
\[ \underbrace{\hspace{3.0cm}}_{n_1}\qquad \underbrace{\hspace{3.0cm}}_{n_2} \]

(where \(\times\) denotes the Kronecker product);

b)
\[ \overset{1}{\circ}-\circ-\cdots-\overset{1}{\circ}\to SO(n(n+2)); \]
\[ \underbrace{\hspace{3.0cm}}_{n} \]

c)
\[ \circ-\circ-\overset{1}{\circ}-\circ-\circ\to Sp(20); \]

d)
\[ \circ-\circ-\begin{matrix}\circ\\[-2pt] |\\[-2pt] \overset{1}{\circ}\end{matrix}-\circ-\circ\to SO(78). \]

Below the Poincaré polynomials of all these spaces are obtained, except for cases a), b).

Let us note that for all symmetric homogeneous spaces \(\mathfrak G/\mathfrak U\), \(\mathfrak U\) is always in normal position in \(\mathfrak G\) (see \((^4)\)). The Poincaré polynomials of all irreducible symmetric spaces \(\mathfrak G/\mathfrak U\) were obtained by Takeuchi \((^2)\).

\(4^\circ\). Consider \(n\) variables \(x_1,x_2,\ldots,x_r\) and the polynomials
\[ p_\alpha=\sum_{i=1}^{n}x_i^\alpha \qquad (\alpha\text{ an integer } >0); \]
then
\[ p_\alpha=Q_\alpha(p_1,p_2,\ldots,p_n). \]
If in \(Q_{ms}\) \((m\text{ an integer } >0;\ 0<s\text{ an integer }<n)\) all arguments are set equal to 0 except for \(p_s,p_{2s},\ldots,p_{i_0s}\) \((i_0=[n/s])\), then the expression obtained will be denoted by
\[ Q_{ms}^{s}(p_s,p_{2s},\ldots,p_{i_0s}). \]
Then the Poincaré polynomials of the preceding spaces have the form:

A. The case \(\mathfrak U\) is not in normal position in \(\mathfrak G\) (see \(3^\circ\)) (for cases a), b) the Poincaré polynomial for large \(n_1,n_2,n\) has not yet been obtained):

c)
\[ P(\mathfrak G/\mathfrak U,t) =(1+t^6+t^{10}+t^{12}+t^{18}+t^{25}+t^{31}+t^{33}+t^{37}+t^{43}) \]
\[ \times(1+t^{27})(1+t^{31})(1+t^{35})(1+t^{39}); \]

d)
\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1+t^9)(1+t^{17})}{(1+t^{19})(1+t^{27})(1+t^{35})} (1+t^{10}+t^{18}+t^{37}+t^{45}+t^{55}). \]

B. \(\mathfrak U\) is in normal position in \(\mathfrak G\) (but is not wholly homologous to 0 in \(\mathfrak G\)):

a)
\[ \circ-\circ-\cdots-\circ \to SU\left(\frac{n(n-1)}{2}\right),\qquad n=2^{s-1}; \]
\[ \underbrace{\hspace{3.0cm}}_{n-1} \]

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1-t^{2ps})}{(1-t^{2s})} \frac{(1+t^{2s-1})}{(1+t^{2ps-1})}, \]

where \(p\) is determined as follows: put
\[ A_1=1,\qquad A_i=-\frac{1}{2n-2^{is}}\sum_{\alpha=1}^{i-1}C_{is}^{\alpha s}A_\alpha A_{i-\alpha} \qquad \left(i=1,2,\ldots,i_0=\left[\frac ns\right]\right); \]
\[ A_m=Q_{ms}^{s}(A_1,A_2,\ldots,A_{i_0}). \]
Then \(p\) is the smallest integer \(>i_0\) such that
\[ (2n-2^{ps})A_p+\sum_{\alpha=1}^{p-1}C_{ps}^{\alpha s}A_\alpha A_{p-\alpha}\ne0. \]

b)
\[ \bullet-\overset{1}{\bullet}-\cdots-\bullet=\circ\to SO((n-1)(2n+1)),\qquad n=2^{2s-2}; \]
\[ \underbrace{\hspace{3.0cm}}_{n} \]

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1-t^{4ps})}{(1-t^{4s})} \frac{(1+t^{4s-1})}{(1+t^{4ps-1})}, \]

where \(p\) is defined as follows: put

\[ A_1=1,\qquad A_i=\frac{-1}{2n-2^{2is-1}}\sum_{\alpha=1}^{i-1} C_{2is}^{2\alpha s}A_\alpha A_{i-\alpha} \quad \left(i=1,2,\ldots,i_0=\left[\frac ns\right]\right); \]

\[ A_m=Q_{ms}^{s}(A_1,A_2,\ldots,A_{i_0}). \]

Then \(p\) is the least integer \(>i_0\) such that

\[ (2n-2^{2ps-1})A_p+\sum_{\alpha=1}^{p-1} C_{2ps}^{2\alpha s}A_\alpha A_{p-\alpha}\ne0. \]

c) \(\circ-\overset{1}{\circ}-\circ-\cdots-\circ\!\begin{matrix}/\circ\\ \backslash\circ\end{matrix}\ \to SO(n(2n-1)):\)

\[ \underbrace{\hspace{3.2cm}}_{n} \]

1) \(n\ne 2^{2s-2}\),

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1+t^{2n-1})}{(1+t^{4n-1})}(1+t^{2n}); \]

2) \(n=2^{2s-2}\),

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1+t^{2n-1})}{(1+t^{4n-1})} \frac{(1+t^{4s-1})}{(1+t^{4ps-1})} (1+t^{2n})\frac{1+t^{4ps}}{1-t^{4s}}, \]

where \(p\) is defined as in case b).

d) \(\overset{1}{\circ}\times\overset{1}{\circ}-\circ-\cdots-\circ=\bullet\to Sp(4n+2);\)

\[ \underbrace{\hspace{3.0cm}}_{n} \]

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1-t^{4p})}{(1-t^4)} \frac{(1+t^3)}{(1+t^{4p-1})}, \]

where \(p\) is defined as follows: put

\[ A_0=n,\qquad A_i=-1-\sum_{\alpha=0}^{i-1} C_{2i}^{2\alpha}A_\alpha \quad (i=1,2,\ldots,n); \]

\[ A_m=Q_m(A_1,A_2,\ldots,A_n). \]

Then \(p\) is the least integer \(>n\) such that

\[ A_p+\sum_{\alpha=0}^{n-1} C_{2p}^{2\alpha}A_\alpha+1\ne0. \]

e) \(\overset{1}{\circ}\times\overset{1}{\circ}-\cdots-\circ\!\begin{matrix}/\circ\\ \backslash\circ\end{matrix}\to Sp(4n);\)

\[ \underbrace{\hspace{3.0cm}}_{n} \]

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)(1+t^{2n-1})}{P(\mathfrak U,t)(1+t^{4n-1})} \frac{(1+t^3)}{(1+t^{4p-1})} (1+t^{2n})\frac{(1-t^{4p})}{(1-t^4)}, \]

where \(p\) is defined as follows: put

\[ A_0=n,\qquad A_i=-\sum_{\alpha=0}^{i-1} C_{2i}^{2\alpha}A_\alpha \quad (i=1,2,\ldots,n); \]

\[ A_m=Q_m(A_1,A_2,\ldots,A_n). \]

Then \(p\) is the least integer \(>n\) such that

\[ A_p+\sum_{\alpha=0}^{p-1} C_{2p}^{2\alpha}A_\alpha\ne0. \]

f) \(\overset{1}{\circ}\times\overset{1}{\bullet}-\bullet-\cdots-\bullet=\circ\to SO(4n),\)

\[ \underbrace{\hspace{3.0cm}}_{n} \]

Set

\[ A_0=n,\quad A_i=-\sum_{\alpha=0}^{i-1} C_{2i}^{2\alpha}A_\alpha,\quad B_i=\sum_{\alpha=0}^{i}(-1)^\alpha C_i^\alpha C_{2i}^{\alpha}A_\alpha \quad (i=1,\ldots,n). \]

Let the function \(R(p_1,p_2,\ldots,p_n)\) express \(x_1x_2\cdots x_n\) in terms of \(p_1,p_2,\ldots,p_n\), \(\left(p_\alpha=\sum_{i=1}^{n}x_i^\alpha\right)\).

1) If \(R(B_1,B_2,\ldots,B_n)\ne0\), then

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1-t^{4n})}{(1-t^4)} \frac{(1+t^3)}{(1+t^{4n-1})}. \]

2) If \(R(B_1,B_2,\ldots,B_n)=0\), then

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1+t^3)}{(1+t^{4p-1})} \frac{(1-t^{4p})}{(1-t^4)}, \]

where \(p\) is determined as follows: set \(A_m=Q_m(A_1,A_2,\ldots,A_n)\); then \(p\) is the least integer \(>n\) such that

\[ A_p+\sum_{\alpha=0}^{p-1} C_{2p}^{2\alpha}A_\alpha\ne0. \]

ж) \(\overset{2}{\circ}-\circ-\cdots-\circ\!\begin{matrix} \circ \\[-0.4ex] | \\[-0.4ex] \circ\end{matrix}\to SO((n+1)(2n-1));\)

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1+t^{2n-1})}{(1+t^{4n-1})}(1+t^{2n}). \]

з) \(\circ-\overset{2}{\circ}-\circ\to Sp(20):\)

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{(1+t^5)}{(1+t^{11})}(1+t^6). \]

\(\overset{1}{\circ}\times\overset{2}{\circ}\to Sp(6):\)

\[ P(\mathfrak G/\mathfrak U,t)=(1+t^4)(1+t^{11}). \]

\(\overset{1}{\circ}\times\overset{1}{\circ}\times\overset{1}{\circ}\to Sp(8):\)

\[ P(\mathfrak G/\mathfrak U,t)=(1+t^4)(1+t^4+t^8)(1+t^{15}). \]

\(\overset{1}{\circ}\times\circ-\overset{1}{\circ}-\circ\to Sp(12):\)

\[ P(\mathfrak G/\mathfrak U,t)=(1+t^4+t^8+t^{12})(1+t^6)(1+t^{19})(1+t^{23}). \]

\(\overset{2}{\circ}-\overset{2}{\circ}\to\overset{1}{\circ}-\circ-\circ-\circ-\circ-\circ:\)

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{1+t^5}{1+t^{11}}(1+t^6). \]

\(\overset{6}{\circ}-\circ+\circ-\overset{6}{\circ}\to\circ-\circ-\circ-\circ-\circ-\overset{1}{\circ}:\)

\[ P(\mathfrak G/\mathfrak U,t)= \frac{P(\mathfrak G,t)}{P(\mathfrak U,t)} \frac{1+t^5}{1+t^{11}}(1+t^6). \]

В. In the remaining cases \(\mathfrak U\) is completely nonhomologous to \(0\) in \(\mathfrak G\).

Moscow State University
named after M. V. Lomonosov

Received
24 VI 1964

CITED LITERATURE

\(^1\) H. S. M. Coxeter, Duke Math. J., 18, No. 4 (1951).
\(^2\) M. Takeuchi, J. Fac. Sci. Univ. Tokyo, Sect. 1, 9, 313 (1962).
\(^3\) O. V. Manturov, a) DAN, 141, No. 4 (1961); b) 141, No. 5 (1961).
\(^4\) J. L. Koszul, Colloque de Topologie (espaces fibrés), Bruxelles, 1950, Paris, 1951.