V. A. BASHEV

REPRESENTATIONS OF THE GROUP \(Z_2 \times Z_2\) OVER A FIELD OF CHARACTERISTIC \(2\) *

(Presented by Academician I. M. Vinogradov on 15 III 1961)

O. Higman \((^2)\) showed that if a Sylow \(p\)-subgroup of a finite group \(G\) is not cyclic, then the group \(G\) has infinitely many inequivalent indecomposable representations by matrices with entries from a field \(k\) of characteristic \(p\). However, the problem of a complete description of all representations is still far from being solved. Even in the simplest situation, when \(G = Z_2 \times Z_2\) and the field \(k\) has characteristic 2, as far as we know, not all representations have been found.

In Section 1 of this paper all indecomposable representations of the group \(Z_2 \times Z_2\) over an algebraically closed field \(k\) of characteristic 2 are found. Further, the set of equivalence classes of representations of a group \(G\) over a field \(k\) forms an associative semiring: addition is induced by the direct sum of representations \(\oplus\), and multiplication by the tensor product of representations \(\otimes\). The minimal ring \(\mathfrak A\) containing this semiring is called the representation ring of the group \(G\) over the field \(k\).

In Section 2 the structure of this ring is studied for \(G = Z_2 \times Z_2\). In particular, it turns out that this ring has an infinite number of generators.

1. Let \(G\) be a \(p\)-group, \(\mathfrak D = k[G]\) the group algebra of the group \(G\) over a field \(k\) of characteristic \(p\); \(I\) the ideal of the algebra \(\mathfrak D\) consisting of elements of the form \(\sum_{\sigma \in G} \alpha_\sigma \sigma\), where \(\sum_{\sigma \in G} \alpha_\sigma = 0\). The \(\mathfrak D\)-module \(I\) is a free \(k\)-module of rank \(p^s - 1\) \((p^s = \operatorname{ord} G)\). Let \(S\) be the ideal consisting of elements \(\gamma \sum_{\sigma \in G} \sigma\), \(\gamma \in k\). As a \(\mathfrak D\)-module the ideal \(S\) is isomorphic to the \(\mathfrak D\)-module \(k\), on which the elements of \(G\) act trivially. The quotient module \(J = \mathfrak D/S\) is a free \(k\)-module of rank \(p^s - 1\). Denote by \(\operatorname{Sp}\) the mapping
   \[ a \mapsto \sum_{\sigma \in G} \sigma a,\quad a \in A. \]

Lemma 1. If \(A\) is an indecomposable \(\mathfrak D\)-module distinct from \(\mathfrak D\), then \(\operatorname{Sp}(A) = (0)\).

The proof of this lemma is given in \((^1)\).

Remark 1. It is clear that all unitary \(\mathfrak D\)-modules are essentially \(I\)-modules, since any \(I\)-module can be made into a \(\mathfrak D\)-module by putting \(e \cdot a = a\), where \(e\) is the identity of the group \(G\) and \(a \in A\). If \(\operatorname{Sp}(A) = (0)\), then the \(I\)-module is in fact an \(N\)-module, where \(N = I/S\). Therefore, by Lemma 1, all indecomposable \(\mathfrak D\)-modules distinct from \(\mathfrak D\) are \(N\)-modules. In the language of representation theory this means that the problem of finding all indecomposable representations of the group \(G\) is equivalent to the problem of finding all representations of the algebra \(N\).

If now \(G = Z_2 \times Z_2\), \(\sigma, \tau\) are its generators and the field \(k\) has characteristic 2, then the ideal \(I\) is generated by the elements \(\sigma - e\), \(\tau - e\), where \(e\) is the identity of the group \(G\), and \(S = I^2\). Therefore \(N = I/I^2\) is an algebra with two generators \(\nu_1\) and \(\nu_2\) and zero multiplication. Denote by \(A'\) the submodule of the module \(A\) consisting of all elements \(a \in A\) such that \(\nu \cdot a = 0\) \((\nu \in N)\), and let \(A'' = A/A'\). Then for elements \(a'' \in A''\) we have \(\nu \cdot a'' = 0\) \((\nu \in N)\).

Lemma 2. If \(A\) and \(B\) are two \(N\)-isomorphic \(N\)-modules, then an \(N\)-isomorphism \(A \xrightarrow{\varphi} B\) induces an \(N\)-isomorphism \(A' \xrightarrow{\varphi'} B'\).

* The results of this work were reported at the seminar on algebraic geometry and homological algebra in Uzhgorod in October 1959.

The proof of Lemma 2 is obvious.

The considerations carried out above in the language of module theory show that, in a suitable basis, every representation of the algebra \(N\) has the form

\[ v_i \to \begin{pmatrix} 0_{nn} & A^{(i)}_{nm}\\ 0_{mn} & 0_{mm} \end{pmatrix}, \qquad i=1,2, \tag{1} \]

where \(A^{(i)}_{nm}\) \((i=1,2)\) are rectangular \((n,m)\)-matrices having no common annihilating vectors; \(0_{nn}, 0_{mn}, 0_{mm}\) are zero matrices; moreover, by Lemma 2, different bases in which the representation has the form (1) are connected by a linear transformation with matrix

\[ Q= \begin{pmatrix} P_{nn} & *\\ 0_{mn} & P_{mm} \end{pmatrix}, \]

where \(P_{nn}, P_{mm}\) are nonsingular square matrices, and \(0_{mn}\) is the zero matrix. Therefore equivalent representations of type (1) have the form

\[ v_i \to \begin{pmatrix} 0_{nn} & P_{nn}^{-1}A^{(i)}_{nm}P_{mm}\\ 0_{nm} & 0_{nn} \end{pmatrix}, \qquad i=1,2. \]

Thus, every representation of the algebra \(N\) is determined by a pair of rectangular matrices \(A^{(i)}_{nm}\) \((i=1,2)\) having no common annihilating vector, and equivalent representations are determined by a pair of matrices \(A^{\prime(i)}_{nm}\) \((i=1,2)\) such that \(A^{\prime(i)}_{nm}=P_{nn}^{-1}A^{(i)}_{nm}P_{mm}\). The pair of matrices \(A^{\prime(i)}_{nm}\) \((i=1,2)\) is called strictly equivalent to the pair of matrices \(A^{(i)}_{nm}\) \((i=1,2)\).

Thus, the problem of finding all representations of the algebra \(N\) is reduced to the problem of reducing a pair of matrices \(A^{(i)}_{nm}\) \((i=1,2)\), or, what is the same thing, the pencil of matrices \(A_{nm}=A^{(2)}_{nm}+\lambda A^{(1)}_{nm}\), to a strictly equivalent canonical form. Since the pair of matrices \(A^{(i)}_{nm}\) \((i=1,2)\) has no common annihilating vector, there is no linear dependence with constant coefficients among the rows of the pencil \(A_{nm}\). Such a pencil (3) is strictly equivalent to a pencil of the form:

\[ (0_{ng}, A^{(2)}_{n,m-g})+\lambda(0_{ng}, A^{(1)}_{n,m-g}), \tag{2} \]

where the pencil \(A_{n,m-g}=A^{(2)}_{n,m-g}+\lambda A^{(1)}_{n,m-g}\) has the quasidiagonal form

\[ A_{n,m-g}=\{L_{\varepsilon_1},\ldots,L_{\varepsilon_r}; L'_{\eta_1},\ldots,L'_{\eta_s}; N^{(1)},\ldots,N^{(t)}, J_1+\lambda E_1,\ldots, \ldots,J_u+\lambda E_u\}. \]

Here

\[ L_{\varepsilon_i}= \left( \begin{array}{ccccccc} \lambda & 1 & 0 & \cdots & \cdots & 0\\ 0 & \lambda & 1 & 0 & \cdots & \cdots\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & \cdot & \cdot & \cdot & \lambda & 1 & 0\\ 0 & 0 & \cdot & \cdot & 0 & \lambda & 1 \end{array} \right\}\varepsilon_i, \]

\[ L'_{\eta_i}= \left( \begin{array}{cccccc} \lambda & 0 & 0 & \cdots & \cdots & 0\\ 1 & \lambda & 0 & \cdots & \cdots & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \lambda & 0\\ 0 & \cdot & \cdot & 1 & \lambda\\ 0 & 0 & \cdot & \cdot & 0 & 1 \end{array} \right\}\eta_i+1, \]

\[ N^{(i)}= \left( \begin{array}{cccccc} 1 & \lambda & 0 & \cdots & \cdots & 0\\ 0 & 1 & \lambda & \cdots & \cdots & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \lambda & 0\\ 0 & \cdot & \cdot & 1 & \lambda\\ 0 & 0 & \cdot & \cdot & 0 & 1 \end{array} \right\}\nu_i, \]

\[ J_i= \left( \begin{array}{cccccc} \lambda_i & 1 & 0 & \cdots & \cdots & 0\\ 0 & \lambda_i & 1 & \cdots & \cdots & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot & 0\\ 0 & \cdot & \cdot & \lambda_i & 1\\ 0 & 0 & \cdot & \cdot & 0 & \lambda_i \end{array} \right\}\mu_i. \]

At the same time it is assumed that the entries of the matrices of the pencil \(A_{nm}\) lie in the algebraically closed field \(k\).

We shall call a pencil \(A\), among whose rows there is no linear dependence with constant coefficients, decomposable if it is strictly equivalent to a pencil of the form (2) or to a pencil of quasidiagonal form \(\{A_1,A_2\}\). The representation (1) of the algebra \(N\) corresponding to the pencil \(A\) will be denoted by \(d(A)\).

Lemma 3. The representation \(d(A)\) is decomposable if and only if the corresponding pencil \(A\) is decomposable.

The proof of this lemma presents no difficulties.

It remains to note that from the uniqueness of the canonical form of a pencil and from Lemma 3 it follows that the representations \(d(L_{\varepsilon_i})\), \(d(L'_{\eta_i})\), \(d(N^{(i)})\), \(d(J_i+\lambda E_i)\) of the algebra \(N\) are indecomposable and that, together with the zero representation, they exhaust all indecomposable representations of the algebra \(N\). Therefore, taking Remark 1 into account, the following may be regarded as proved:

Proposition 1. All indecomposable nonequivalent representations of the group \(Z_2\times Z_2\) over an algebraically closed field \(k\) of characteristic \(2\) are exhausted by the following representations:

1) The unit representation
\[ \varepsilon(\sigma)=1,\qquad \varepsilon(\tau)=1. \]

2) Representations of order \(2s+1,\ s=1,2,\ldots,\)
\[ I_{2s+1}(\sigma)= \begin{pmatrix} E_{s+1} & L'_s(0)+L'_s(1)\\ & E_s \end{pmatrix}, \qquad I_{2s+1}(\tau)= \begin{pmatrix} E_{s+1} & L'_s(0)\\ & E_s \end{pmatrix}. \]
\[ J_{2s+1}(\sigma)= \begin{pmatrix} E_s & L_s(0)+L_s(1)\\ & E_{s+1} \end{pmatrix}, \qquad J_{2s+1}(\tau)= \begin{pmatrix} E_s & L_s(0)\\ & E_{s+1} \end{pmatrix}. \]

3) Representations of order \(2s,\ s=1,2,\ldots,\)
\[ \Delta_{2s}(\lambda)(\sigma)= \begin{pmatrix} E_s & E_s\\ & E_s \end{pmatrix}, \qquad \Delta_{2s}(\lambda)(\tau)= \begin{pmatrix} E_s & J_s(\lambda)\\ & E_s \end{pmatrix}, \qquad \lambda\in k,\quad s=1,2,\ldots, \]
\[ \Delta_{2s}(\infty)(\sigma)= \begin{pmatrix} E_s & J_s(0)\\ & E_s \end{pmatrix}, \qquad \Delta_{2s}(\infty)(\tau)= \begin{pmatrix} E_s & E_s\\ & E_s \end{pmatrix}. \]

4) The regular representation \(R\) of order \(4\).

1. As before, \(G=Z_2\times Z_2\); \(k\) is a field of characteristic \(2\). The \(k\)-module \(I\) in this case is a free \(k\)-module of rank 3. The representation of dimension 3 of the group \(G\) corresponding to this module will also be denoted by \(I\). \(J\) is also a free \(k\)-module of rank 3. The corresponding representation of dimension 3 will likewise be denoted by \(J\).

Remark 2. One can verify that the representation \(I\) is equivalent to the representation \(I_3\), and the representation \(J\) is equivalent to the representation \(J_3\), obtained by us in Proposition 1. In [1] it is proved that the representation \(I\) is dual to the representation \(J\). It turns out that the representation \(I_{2s+1}\) is also dual to the representation \(J_{2s+1}\). Obviously, \(R\) is dual to itself.

In the subsequent computations the following will be useful.

Lemma 4. Let \(G\) be a \(p\)-group, and let \(k\) be a field of characteristic \(p\). Then the regular representation \(R\) of the group \(G\) is contained as a direct summand in a representation \(D\) as many times as the rank of the matrix
\[ \operatorname{Sp} D=\sum_{\sigma\in G}D(\sigma). \]

This lemma is easily proved with the aid of Lemma 1.

Let us now proceed to the computation of the representation ring \(\mathfrak A\) of the group \(G\). Obviously, \(\varepsilon\) is the identity of this ring. Moreover, it is known that
\[ D\otimes R= \underbrace{R\oplus\cdots\oplus R}_{m\ \text{times}} =mR, \]
where \(m\) is the dimension of the representation \(D\).

Proposition 2.
\[ I_{2s+1}\otimes I_{2t+1}=I_{2(s+t)+1}\oplus stR, \]
\[ J_{2s+1}\otimes J_{2t+1}=J_{2(s+t)+1}\oplus stR, \]
\[ I_{2s+1}\otimes J_{2t+1}=I_{2(s-t)+1}\oplus (st+t)R,\quad \text{if } s>t. \]
\[ I_{2s+1}\otimes J_{2s+1}=\varepsilon\oplus (s^2+s)R, \]
\[ I_{2s+1}\otimes J_{2t+1}=J_{2(t-s)+1}\oplus (st+s)R,\quad \text{if } s<t. \]

The proof is based on Lemma 4, Proposition 1, and the results of [1].

Lemma 5.

\[ \Delta_{2n}(\lambda)\otimes \Delta_{2m}(\mu)=nmR,\qquad \text{if } \lambda\ne\mu, \]

\[ \Delta_{2n}(\lambda)\otimes \Delta_{2m}(\lambda) =(nm-\min(m,n))R\oplus D_{4\min(m,n)}, \]

where \(D_{4\min(m,n)}\) is a representation of the group \(G\) of dimension \(4\min(m,n)\), not containing \(R\).

This lemma is easily proved with the aid of Lemma 4.

Proposition 3.

\[ I_{2s+1}\otimes \Delta_{2t}(\lambda)=\Delta_{2t}(\lambda)\oplus stR, \]

\[ J_{2s+1}\otimes \Delta_{2t}(\lambda)=\Delta_{2t}(\lambda)\oplus stR. \]

The proof is based on Lemmas 4, 5, Propositions 1, 2, and the results of [1].

The following lemma, which we shall state without proof, will be needed later.

Lemma 6. Let \(\mathfrak A\) be an associative, commutative ring with generators \(x_s\) \((s=1,2,\ldots;\ x_0=0)\), on which an integer-valued linear functional \(f\) is defined such that \(f(x_s)=s,\ f(x_s\cdot x_t)=2\min(s,t)\). Then \(x_m\cdot x_n=2x_{\min(m,n)}\), if \(m\ne n\).

Proposition 4.

\[ \Delta_{2n}(\lambda)\otimes \Delta_{2m}(\mu)=nmR,\qquad \lambda\ne\mu. \tag{3} \]

For \(\lambda\ne 0,1,\infty\):

\[ \Delta_{2n}(\lambda)\otimes \Delta_{2m}(\lambda) =(nm-\min(m,n))R\oplus 2\Delta_{2\min(m,n)}(\lambda),\quad m\ne n; \tag{4} \]

\[ \Delta_{2n}(\lambda)\otimes \Delta_{2n}(\lambda)= \begin{cases} (n^2-n)R\oplus 2\Delta_{2n}(\lambda),& n\equiv 0\ (2),\\ (n^2-n)R\oplus \Delta_{2(n-1)}\oplus \Delta_{2(n+1)},& n\equiv 1\ (2); \end{cases} \tag{5} \]

For \(\lambda=0,1,\infty\):

\[ \Delta_{2n}(\lambda)\otimes \Delta_{2m}(\lambda) =(nm-\min(m,n))R\oplus 2\Delta_{2\min(m,n)}(\lambda). \tag{6} \]

Proof. Formula (3) was established by us in Lemma 5. It was also established there that \(R\) is contained in \(\Delta_{2n}(\lambda)\otimes \Delta_{2m}(\lambda)\) exactly \((nm-\min(n,m))\) times. Let \(\mathfrak A\) be the representation ring of the group \(G\), \(W\) the ideal generated by the regular representation \(R\), and \(\mathfrak A^*=\mathfrak A/W\). Using the results of Proposition 3 and Lemma 5, we find that, for fixed \(\lambda\), the linear subspace \(\mathfrak B_\lambda^*\), generated by the classes \(x_s=\{\Delta_{2s}(\lambda)\bmod W\}\), forms a subring of the ring \(\mathfrak A^*\). Define on this ring an integer-valued linear functional \(f\), putting \(f(x_s)=s\). According to Lemma 5, \(f(x_s\cdot x_t)=2\min(s,t)\), \(s\ne t\). To obtain (4), it remains to apply Lemma 6 to the ring \(\mathfrak B_\lambda^*\). For the proof of relations (5) and (6) it is necessary to carry out rather lengthy computations, essentially using the structure of the representations \(\Delta_{2s}(\lambda)\). We omit these computations. This completes the proof of Proposition 4.

It remains to say that Propositions 2, 3, and 4 completely determine the structure of the ring \(\mathfrak A\). In particular, it is clear that this ring has no finite number of generators.

The present work was carried out under the supervision of I. R. Shafarevich, to whom the author expresses sincere gratitude.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
9 III 1961

REFERENCES

1. D. K. Fadeev, Z. I. Borevich, Vestn. LGU, No. 7, 72 (1959).
2. D. G. Higman, Duke Math. J., 21, No. 2, 377 (1954).
3. F. R. Gantmacher, Theory of Matrices, Moscow, 1954, Ch. XII, § 4.