GERT ALMKVIST’S GENERALIZATION OF A MISTAKE OF BOURBAKI ^{1}

Doron Zeilberger ^{2}

Dedicated to the memory of Emil Grosswald

Appeared in: Contemporary Mathematics 143(1993), 609-612.

Theorem([A1]): Let f be an n × n matrix whose entries are drawn from a commutative ring A. f^{m+1} = 0 implies (T r(f))^{nm+1} = 0 ^{3} ^{4}

Proof([A1]): Let e_{1}, ..., e_{n} be the standard basis of A^{n}, and think of f as a linear transformation defined on the basis elements by

f(e_{i}) = ∑_{j=1}^{n} f_{i,j}e_{j} , and extended by linearity .

Consider the algebra ΛA^{n} ^{5} generated by the ”indeterminates” e_{1}, ..., e_{n} with coefficients in the ring A and product ∧ defined by associativity, distributivity, and the commutation relations e_{i} ∧ e_{j} = −e_{j} ∧ e_{i}. In particular, e_{i} ∧ e_{i} = 0, for any e_{i}, and hence any ”monomial” e_{i_{1}} ∧ ... ∧ e_{i_{r}} that has i_{a} = i_{b} for some 1 <= a < b <= r (i.e. has repeated factors) must vanish.

It is easily seen that

(T r f)(e_{1} ∧ ... ∧ e_{n}) = ∑_{i=1}^{n} e_{1} ∧ ... ∧ f(e_{i}) ∧ ... ∧ e_{n} ,  (∗)

1 A long time ago, Gert Almkvist told me that one day he will tell me why he is known among his friends and colleagues as ”the guy who generalized a mistake of Bourbaki”. He finally fulfilled his promise, and I was so excited about his beautiful proof, that I decided to share it with the rest of the mathematical world, especially since his proof is buried inside a very technical paper([A1]). This mistake lead Almkvist to the foundation of the algebraic K-theory of endomorphisms in [A2], that contains the result that there are no other additive invariants of an endomorphism of a finitely generated projective module (over any commutative ring) than the characteristic polynomial (and the class of the module in K(0, A)).

2 Department of Mathematics, Temple University, Philadelphia, PA19122. Supported in part by NSF grant DMS8800663. I wish to thank Martin Lorenz, Shmuel Rosset, and the referee, for very helpful remarks.

3 This generalizes, and corrects the following mistake in Bourbaki’s [B] p. 156, ex. 13d: ”if f is a 2 × 2 matrix then f^{2} = 0 implies (T r f)^{4} = 0 and 4 is the smallest integer with this property”. Almkvist’s theorem shows that 4 should be replaced by 3.

4 Note that if A is an algebraically closed field, then the result follows immediately from the Jordan canonical form, since a nilpotent matrix is strictly upper-triangular w.r.t to some basis, and hence trace-less. However Almkvist’s result is true for any commutative ring. Note also that if f is diagonal (or even upper-triangular), then Almkvist’s theorem follows from the multinomial theorem. Note also that nm + 1 is best possible, since if f is the diagonal matrix diag(x_{1}, ..., x_{n}), with x_{i}^{m+1} = 0 but , x_{i}^{m} ≠ 0, (i = 1, ..., n) , then (T r f)^{nm} = ((nm)!/m!^{n})(x_{1}...x_{n})^{m}, which in general is not zero.

5 The so-called exterior (or Grassmann) algebra and it follows by anti-symmetry and multi-linearity that for any vectors b_{1}, ..., b_{n} in A^{n} (i.e. the b’s are linear combinations of the e’s, with coefficients in A):

(T r f)(b_{1} ∧ ... ∧ b_{n}) = ∑_{i=1}^{n} b_{1} ∧ ... ∧ f(b_{i}) ∧ ... ∧ b_{n} .  (∗∗)

By starting with (*), and iterating according to (**), it follows that for any positive integer N, (T r f)^{N}(e_{1} ∧ ... ∧ e_{n}) is a linear combination of terms of the form f^{α_{1}}e_{1} ∧ ...∧ f^{α_{n}}e_{n}, with α_{1} + ... + α_{n} = N. ^{6} But if N = nm + 1, each such term is 0, since at least one of the α_{i} is bound to be > m. QED

Almkvist[A1] used the same idea to prove a more general theorem. ^{7} Other applications of the method include det(fg) = det(f)det(g), ^{8} Jacobi’s det(exp(f)) = exp(T r(f)), ^{9} the Cayley-

6 More precisely, we have the ”multinomial” theorem

(T r f)^{N}(e_{1} ∧ ... ∧ e_{n}) = ∑_{α_{1}+...+α_{n}=N} \frac{N!}{α_{1}!...α_{n}!}f^{α_{1}}e_{1} ∧ ... ∧ f^{α_{n}}e_{n ,}

since (**) says that with the correspondence

f^{α_{1}}e_{1} ∧ ... ∧ f^{α_{n}}e_{n →}x_{1}^{α_{1}}...x_{n}^{α_{n}} ,

multiplying from the left by T r f is like multiplying by (x_{1} + ... + x_{n}). 7 (T r f) can be thought of as the sum of all the 1 × 1 principal minors of f. Let a_{r}(f) be the sum of

all the r × r principal minors, then Almkvist’s more general theorem says that a_{1}^{nu_{1}} · ·· a_{n}^{nu_{n}} = 0, whenever nu_{1} +2nu_{2} +...+nnu_{n} > mn. To prove this Almkvist first observed that (*) generalizes to

a_{r}(f)(e_{1} ∧ ... ∧ e_{n}) = ∑_{1<=i_{1}<...<i_{r}<=n} e_{1} ∧ ... ∧ f(e_{i_{1}}) ∧ ... ∧ f(e_{i_{r}}) ∧ ... ∧ e_{n} ,  (∗′)

and the obvious generalization of (**), with the e_{i} in (*’) replaced by general b_{i} also holds by multilinearity, so by iterating, a_{1}^{nu_{1}} · ·· a_{n}^{nu_{n}} is an integer-linear combination of terms of the form f^{α_{1}}e_{1} ∧ ...∧ f^{α_{n}}e_{n}, with α_{1} + ... + α_{n} = nm + 1, so must vanish, just as before.

8

det(fg)(e_{1} ∧ ... ∧ e_{n}) = fg(e_{1}) ∧ ... ∧ fg(e_{n}) = det(f)g(e_{1}) ∧ ... ∧ g(e_{n}) = det(f)det(g)e_{1} ∧ ... ∧ e_{n} .

9

exp(T r(f))(e_{1}∧...∧e_{n}) = ∑_{r=0}^{∞}(1/r!)T r(f)^{r}(e_{1}∧...∧e_{n}) = ∑_{r=0}^{∞}(1/r!) ∑_{α_{1}+...α_{n}=r} \frac{r!}{α_{1}!...α_{n}!}f^{α_{1}}e_{1} ∧...∧f^{α_{n}}e_{n} =

Hamilton-type theorem (a_{r} as in footnote 6) ^{10}

∑_{r=0}^{m}(−1)^{r}a_{r}(f)T r(f^{m−r}) = 0 ,  (C − H)

and Rosset’s [R] beautiful proof of the Amitsur-Levitski theorem. ^{11}

∑_{α_{1}>=0,...,α_{n}>=0} \frac{1}{α_{1}!...α_{n}!}f^{α_{1}}e_{1} ∧ ... ∧ f^{α_{n}}e_{n} = exp(f)e_{1} ∧ ... ∧ exp(f)e_{n} = det(exp(f))e_{1} ∧ ... ∧ e_{n} .

Jacobi’s identity can be used to give another proof of Almkvist’s theorem (but not of its generalization in footnote 6): Since exp(tf) is a polynomial of degree m in t, det(exp(tf)) is a polynomial of degree mn, and hence the coefficient of t^{mn+1} in exp(tT r(f)) vanishes.

10 Going back to the correspondence with multinomials of footnote 5, (*’) and its ”b-analog” imply that multiplying from the left by a_{r}(f) corresponds to multiplying by the elementary symmetric function

E_{r}(x_{1}, ..., e_{n}) = ∑_{1<=i_{1}<...<i_{r}<=n} x_{i_{1}}...x_{i_{r}} ,

and multiplying by T r(f^{r}), corresponds to multiplying by the power functions

P_{r}(x_{1}, ..., x_{n}) = ∑_{i=1}^{n} x_{i}^{r} .

It follows that any polynomial identity in T r(f^{r}) and a_{r}(f), holds if it holds when T r(f^{r}) is replaced by P_{r} and a_{r}(f) is replaced by E_{r}. But then the identity becomes Newton’s celebrated identities (see [Z] for a combinatorial proof). In fact, Newton’s identities are the special case when f is upper-triangular. When A is an algebraically closed field, the general identity follows from the Jordan canonical form, but the above argument is simpler, and holds for any commutative ring A.

11 The Amitsur-Levitski identity (see [R]) states that if f_{1}, ..., f_{2n} are n × n matrices whose entries belong to a commutative ring A, then

∑_{πϵS_{2n}} sgn(π)f_{π(1)}...f_{π(2n)} = 0.

Rosset’s [R] ingenious proof goes as follows. Let f be the following matrix, (whose entries are linear combinations of e_{i})

f := (f_{1})e_{1} + ... + (f_{2n})e_{2n ,}

(Do not confuse (f_{1})e_{1} with f_{1}(e_{1}). The former is a matrix with elements that are (linear) exterior REFERENCES

[A1] Almkvist, Gert, Endomorphisms of finitely many projective modules over a commutative ring, Arkiv fo:r matematik 11(1973), 263-301.

[A2] Almkvist, G., The Grothendieck ring of the category of endomorphisms, J. Alg. 28 (1974), 375-388.

[B] Bourbaki, Nicolas, Algèbre commutative, ch. I. and II., (Hermann, Paris, 1961).

[R] Rosset, Shmuel, New proof of the Amitsur-Levitski identity, Israel J. Math. 23(1976), 187-188.

[Z] Zeilberger, Doron, A combinatorial proof of Newton’s identities, Discrete Math. 49(1984) ,319.

forms, while the latter is an exterior linear form itself.) The Amitsur-Levitski identity is equivalent to the statement that f^{2n} = 0. Rosset now claim that T r(f^{2r}) = 0, for r > 0. Since T r(gh) = T r(hg), it follows that the trace of any product of matrices is not changed when the leftmost matrix is moved to the extreme right. The above operation, when applied to a product of matrices of even length, takes even permutations into odd ones (since we multiplied the underlying permutation by a cycle of even length, which has sign −1). The coefficient of a typical term e_{i_{1}} ∧ ... ∧ e_{i_{2}}r in f^{2r} is

∑_{πϵS_{2r}} sgn(π)f_{i_{π(1)}}...f_{i_{π(2r)}} .

It is clear that its trace is zero, since the terms can be arranged in mutually canceling pairs, as explained above.

Now, the entries of f do not commute, but the entries of f^{2} do! They are linear combinations of pairs e_{i} ∧ e_{j}, and since (−1)^{2} = +1, all such pairs mutually commute. Now since T r((f^{2})^{r}) = 0, r = 1, 2, ..., n, it follows from (C-H) that a_{r}(f^{2}) = 0, r = 1, ..., n, which implies (f^{2})^{n} = 0 by the Cayley-Hamilton theorem.