Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions
aa r X i v : . [ m a t h . R A ] J u l STRUCTURE THEORY FOR THE GROUP ALGEBRA OFTHE SYMMETRIC GROUP, WITH APPLICATIONS TOPOLYNOMIAL IDENTITIES FOR THE OCTONIONS
MURRAY R. BREMNER, SARA MADARIAGA, AND LUIZ A. PERESI
To our colleague Irvin Roy Hentzel on his 71st birthday
Abstract.
In part 1, we review the structure theory of F S n , the group algebraof the symmetric group S n over a field of characteristic 0. We define the images ψ ( E λij ) of the matrix units E λij (1 ≤ i, j ≤ d λ ), where d λ is the number ofstandard tableaux of shape λ , and obtain an explicit construction of Young’sisomorphism ψ : L λ M d λ ( F ) → F S n . We then present Clifton’s algorithm forthe construction of the representation matrices R λ ( p ) ∈ M d λ ( F ) for all p ∈ S n ,and obtain the reverse isomorphism φ : F S n → L λ M d λ ( F ).In part 2, we apply the structure theory of F S n to the study of multilinearpolynomial identities of degree n ≤ O of octonions over afield of characteristic 0. We compare our results with earlier work of Racine,Hentzel & Peresi, and Shestakov & Zhukavets on the identities of degree n ≤ ≤ Contents
1. Structure theory for the group algebra of the symmetric group 21.1. Young diagrams and tableaux 21.2. Horizontal and vertical permutations 41.3. Row and column intersections 51.4. Symmetric and alternating sums 61.5. Idempotents and orthogonality in the group algebra 71.6. Two-sided ideals in the group algebra 91.7. Matrix units in the group algebra 111.8. Clifton’s theorem on representation matrices 132. Computational methods for studying polynomial identities 162.1. Historical background 162.2. Multilinear polynomial identities satisfied by an algebra 172.3. Consequences of polynomial identities in higher degrees 192.4. Representations of S n and multilinear identities in degree n T -ideals 23 Mathematics Subject Classification.
Primary 20C30. Secondary 16K20, 16S34, 17A30,17A75, 17D05, 17-08, 20B30, 20C40.
Key words and phrases.
Symmetric group, representation theory, group algebra, Youngtableaux, idempotents, matrix units, two-sided ideals, Wedderburn decomposition, Clifton’s algo-rithm, polynomial identities, nonassociative algebra, octonions, computer algebra.
Structure theory for the group algebra of the symmetric group
In this first part, we study the structure of the group algebra F S n of the sym-metric group S n on n letters. As a vector space over F , F S n has basis { σ | σ ∈ S n } ,and the associative multiplication is defined on basis elements by the product in S n and extended bilinearly. We assume throughout that F is a field of characteristic 0.By the classical structure theory of associative algebras, we know that F S n issemisimple, and hence isomorphic to the direct sum of full matrix algebras withentries in division algebras over F . In fact, each of these division algebras is iso-morphic to F , and the Wedderburn decomposition is given by two isomorphisms,(W) φ : F S n −→ M λ M d λ ( F ) , ψ : M λ M d λ ( F ) −→ F S n , where the sum is over all partitions λ of n , and d λ is the dimension of the irreduciblerepresentation of S n corresponding to λ .The matrices obtained by restricting φ to S n , and taking the component of φ forpartition λ , have entries in { , ± } and form the natural representation of S n . Wewill show how to efficiently compute these matrices for all λ and all p ∈ S n .Each matrix algebra M d λ ( F ) has a basis of matrix units E λij for i, j = 1 , . . . , d λ which multiply according to the standard relations, E λij E µkℓ = δ λµ δ jk E λiℓ . The isomorphism ψ produces elements ψ ( E λij ) in F S n which obey the same equa-tions. We show how to calculate these elements of F S n .None of the material in this first part is original. We compiled the results frommany sources, and attempted to make the terminology more contemporary andthe notation simpler and more consistent. The structure theory of F S n was orig-inal worked out by Young [42]. The proofs in Young’s papers were simplified byRutherford [36], and the theory was reformulated in more modern terminologyand notation by Boerner [4], following suggestions by von Neumann and van derWaerden [41]. A substantial simplification of the algorithms for computing the ma-trices in the natural representation (the isomorphism φ ) was introduced by Clifton[13, 14]. Our exposition is based on the Ph.D. thesis of Bondari [5, 6].1.1. Young diagrams and tableaux.
We start by giving the definitions andelementary properties of the basic objects in the theory. The symmetric group S n is the group of all permutations of the set { , . . . , n } . We write λ ⊢ n to indicatethat λ is a partition of n ; that is, λ = ( n , . . . , n k ) where n = n + · · · + n k and n ≥ · · · ≥ n k ≥
1. If n ≤ λ = n · · · n k . Definition 1.1.
The
Young diagram Y λ of the partition λ = ( n , . . . , n k ) con-sists of k left-justified rows of empty square boxes where row i contains n i boxes. HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 3
Example 1.2.
Young diagrams for some partitions of n = 9: Y = Y = Y = Definition 1.3.
Suppose that λ = ( n , . . . , n k ) and λ ′ = ( n ′ , . . . , n ′ ℓ ) are partitionsof n . We say that λ ≺ λ ′ (equivalently, Y λ ≺ Y λ ′ ) if and only if either n < n ′ orthere exists i ≥ n = n ′ , . . . , n i = n ′ i but n i +1 < n ′ i +1 . Example 1.4.
The seven Young diagrams for n = 5 in decreasing order: Definition 1.5. A Young tableau T λ of shape λ where λ ⊢ n consists of abijective assignment of the numbers 1 , . . . , n to the boxes in the Young diagram Y λ . The number in row i and column j will be denoted T ( i, j ). The sequence ofnumbers from left to right in row i will be denoted T ( i, − ); the sequence of numbersfrom top to bottom in column j will be denoted T ( − , j ). A tableau is standard ifall the sequences T ( i, − ) and T ( − , j ) are increasing. Remark 1.6.
The number d λ of standard tableaux for the Young diagram Y λ isgiven by the hook formula, d λ = n ! Q i,j | h ij | , where | h ij | is the number of boxes in the hook with corner at position ( i, j ): h ij = { ( i, j ′ ) | j ≤ j ′ } ∪ { ( i ′ , j ) | i ≤ i ′ } . Another way to write this formula which is easier to implement on a computer is d λ = n ! Q i Given two tableaux T and T ′ of shape λ ⊢ n , let i be the least rowindex for which T ( i, − ) = T ′ ( i, − ), and let j be the least column index for which T ( i, j ) = T ′ ( i, j ). The lexicographical order (lex order) on tableaux is definedby T ≺ T ′ if and only if T ( i, j ) < T ′ ( i, j ). Example 1.8. The standard tableaux for n = 5, λ = 32 in lex order:1 2 34 5 1 2 43 5 1 2 53 4 1 3 42 5 1 3 52 4 Definition 1.9. For each partition λ ⊢ n , the group S n acts on the tableaux ofshape λ by permuting the numbers in the boxes. For p ∈ S n and tableau T , theresult will be denoted pT : that is, if T ( i, j ) = x then ( pT )( i, j ) = px . BREMNER, MADARIAGA, AND PERESI Horizontal and vertical permutations. Each tableau of shape λ ⊢ n de-termines certain subgroups of S n which play an essential role in the theory. Definition 1.10. Given a tableau T of shape λ = ( n , . . . , n k ) ⊢ n , we write G H ( T ) for the subgroup of S n consisting of all horizontal permutations for T .These are the permutations h ∈ S n which leave the rows fixed as sets: for all i = 1 , . . . , k , if x ∈ T ( i, − ) then hx ∈ T ( i, − ). Similarly, the subgroup G V ( T ) of vertical permutations of T consists of all permutations v ∈ S n which leave thecolumns fixed as sets: for all j = 1 , . . . , n , if x ∈ T ( − , j ) then vx ∈ T ( − , j ). Remark 1.11. If we regard the rows T ( i, − ) and columns T ( − , j ) as sets, then G H ( T ) and G V ( T ) can be defined as direct products: G H ( T ) = k Y i =1 S T ( i, − ) , G V ( T ) = n Y j =1 S T ( − ,j ) , where S X denotes the group of all permutations of the set X . Lemma 1.12. If T is a tableau of shape λ ⊢ n then G H ( T ) ∩ G V ( T ) = { ι } where ι ∈ S n is the identity permutation. It follows that if h, h ′ ∈ G H ( T ) and v, v ′ ∈ G V ( T ) with hv = h ′ v ′ then h = h ′ and v = v ′ .Proof. If hv = h ′ v ′ then ( h ′ ) − h = v ′ v − and so both equal ι . (cid:3) xxki j ℓ T pxpxki j ℓ pT pp − q pqp − Figure 1. Tableaux for the proof of Lemma 1.13 Lemma 1.13. Assume that T is a tableau of shape λ ⊢ n and p ∈ S n .(a) If h ∈ G H ( T ) then php − ∈ G H ( pT ) . Since conjugation by p is invertible, itis a bijection from G H ( T ) to G H ( pT ) .(b) If v ∈ G V ( T ) then pvp − ∈ G V ( pT ) . Since conjugation by p is invertible, itis a bijection from G V ( T ) to G V ( pT ) .Proof. We refer to Figure 1. Suppose the permutation q ∈ S n moves the number x from position ( i, j ) of the tableau T to position ( k, ℓ ); this is represented by thearrow labelled q in the left tableau. Following the lower curved arrow labelled p − ,then the arrow in the left tableau labelled q , and finally the upper curved arrowlabelled p , we see that the permutation pqp − moves x ′ = px from position ( i, j ) ofthe tableau pT to position ( k, ℓ ). This is represented by the arrow labelled pqp − in the right tableau. In particular, if q = h ∈ G H ( T ) then i = k , and so php − isa horizontal permutation for pT . Similarly, if q = v ∈ G V ( T ) then j = ℓ , and so pvp − is a vertical permutation for pT . (cid:3) HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 5 Remark 1.14. The notation hvT indicates that we apply the vertical permutation v ∈ G V ( T ) to T and then apply the horizontal permutation h ∈ G H ( T ) to vT .However, h may not be a horizontal permutation for vT . We can rewrite this usingpermutations which are horizontal or vertical for the tableaux on which they act:we have hvT = ( hvh − ) hT and hvh − is a vertical permutation for hT .1.3. Row and column intersections. The next few results investigate the inter-section T ( i, − ) ∩ T ′ ( − , j ) for tableaux T and T ′ of shapes λ and µ . Proposition 1.15. Assume that λ, µ ⊢ n with Y λ ≻ Y µ . For any tableaux T λ , T µ there exist i, j for which T λ ( i, − ) ∩ T µ ( − , j ) contains at least two numbers. Thusthere exist two numbers in one row of T λ which appear in one column of T µ .Proof. Write λ = ( n , . . . , n k ) and µ = ( n ′ , . . . , n ′ ℓ ). We make the contrary as-sumption that T λ ( i, − ) ∩ T µ ( − , j ) contains at most one number for all 1 ≤ i ≤ k and 1 ≤ j ≤ n ′ . In particular, for i = 1 we see that the n numbers in T λ (1 , − )belong to different columns of T µ , and so n ≤ n ′ . But Y λ ≻ Y µ implies n ≥ n ′ ,and so n = n ′ . The contrary assumption is not affected if we apply a vertical per-mutation to T µ , and so there exists v ∈ G V ( T µ ) for which T λ (1 , − ) = ( vT µ )(1 , − )as sets; these rows contain the same numbers, possibly in different order.We now delete the first rows of T λ and vT µ , obtaining tableaux T λ ′ ≻ T µ ′ where λ ′ , µ ′ are partitions of n − n . Both tableaux contain the numbers { a , . . . , a n − n } ⊂{ , . . . , n } which we can identify with { , . . . , n − n } . Repeating the argument ofthe first paragraph, we see that n = n ′ , . . . , n k = n ′ ℓ ; at the end we must have k = ℓ . This implies that Y λ = Y µ , which is a contradiction. (cid:3) Lemma 1.16. Let T be a tableau of shape λ = ( n , . . . , n k ) ⊢ n . A permutation p ∈ S n has the form p = hv for h ∈ G H ( T ) and v ∈ G V ( T ) if and only if T ( i, − ) ∩ ( pT )( − , j ) contains at most one number for all i = 1 , . . . , k and j = 1 , . . . , n .Proof. Assume that p = hv for some h ∈ G H ( T ) and v ∈ G V ( T ). Following Remark1.14, we have pT = hvT = ( hvh − ) hT where hvh − ∈ G V ( hT ). If x, y are distinctnumbers in the same row of T , then they are in the same row but different columnsof hT ; hence they are in different columns of ( hvh − ) hT = pT .Conversely, assume that T ( i, − ) ∩ ( pT )( − , j ) contains at most one number forall i = 1 , . . . , k and j = 1 , . . . , n . Then the numbers in the first column of pT mustappear in different rows of T . We can apply a horizontal permutation h ∈ G H ( T )so that ( h T )( − , 1) is a permutation of ( pT )( − , pT must appear in different rows of h T and columns j ≥ h T )( − , 1) fixed, we can apply h ∈ G H ( T ) so that( h h T )( − , 2) is a permutation of ( pT )( − , h , h , . . . , h n ∈ G H ( T ) so that every number in hT (where h = h n · · · h ) is inthe same column as in pT . We now apply a vertical permutation v ′ ∈ G V ( hT ) toobtain v ′ hT = pT . By Lemma 1.13, we have v ′ = hvh − for some v ∈ G V ( T ).Therefore pT = v ′ hT = hvh − hT = hvT , as required. (cid:3) Proposition 1.17. Assume that λ = ( n , . . . , n k ) ⊢ n , and let T , . . . , T d λ be thestandard tableaux of shape λ in lex order. If r > s then there exist i ∈ { , . . . , k } and j ∈ { , . . . , n } such that T r ( − , j ) ∩ T s ( i, − ) contains at least two elements.Proof. Let ( i ′ , j ′ ) be the first position in which T r and T s have a different number.Let x, y be the numbers in position ( i ′ , j ′ ) in T r , T s respectively. Since r > s we BREMNER, MADARIAGA, AND PERESI T r · · · j ′′ · · · j ′ · · · ... ... ... i ′ · · · z · · · x · · · ... ... ... i ′′ · · · y · · · ... ... T s · · · j ′′ · · · j ′ · · · ... ... ... i ′ · · · z · · · y · · · ... ... ... i ′′ · · · · · · ... ... Figure 2. Diagram for the proof of Proposition 1.17have x > y . In a standard tableau, each number in the first column is the leastnumber that has not appeared in previous rows. Hence j ′ ≥ 2. Suppose that y occurs in position ( i ′′ , j ′′ ) in T r . Since T r and T s are equal up to position ( i ′ , j ′ ),we have two cases: either i ′′ = i ′ and j ′′ > j ′ ( y is in the same row as x but to theright), or i ′′ > i ′ ( y is in a lower row than x ). Since x > y and T r is standard, thefirst case is impossible. In the second case, x > y implies j ′′ < j ′ ( y must be in acolumn to the left of x ). We illustrate this situation with the diagram of Figure 2.Since position ( i ′ , j ′′ ) occurs before ( i ′ , j ′ ), the number z in this position must bethe same in both T r and T s . Hence y, z are the two numbers in the same columnof T r and the same row of T s . (cid:3) Symmetric and alternating sums. We construct special elements of F S n which will be used to define idempotents in the group algebra. Definition 1.18. Given a tableau T of shape λ ⊢ n we define the following elementsof F S n , where ǫ : S n → {± } is the sign homomorphism: H T = X h ∈ G H ( T ) h, V T = X v ∈ G V ( T ) ǫ ( v ) v. (Classically these were called the “positive and negative symmetric groups” for T .) Lemma 1.19. If T is a tableau of shape λ ⊢ n , and h ∈ G H ( T ) , v ∈ G V ( T ) , then hH T = H T = H T h, vV T = ǫ ( v ) V T = V T v. Proof. For a horizontal permutation h , the function G H ( T ) → G H ( T ) sending h ′ hh ′ is a bijection, and similarly for h ′ h ′ h ; this proves the claim for H T .Analogous bijections hold for G V ( T ) and a vertical permutation v , so vV T = X v ′ ∈ G V ( T ) ǫ ( v ′ ) vv ′ = ǫ ( v ) − X v ′ ∈ G V ( T ) ǫ ( v ) ǫ ( v ′ ) vv ′ = ǫ ( v ) X v ′ ∈ G V ( T ) ǫ ( vv ′ ) vv ′ = ǫ ( v ) V T . The proof that ǫ ( v ) V T = V T v is similar. (cid:3) Proposition 1.20. If T is a tableau of shape λ ⊢ n , and p ∈ S n , then H pT = p H T p − , V pT = p V T p − . Proof. This follows from Lemmas 1.13 and 1.19 since ǫ ( p ) = ǫ ( p − ). (cid:3) HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 7 Idempotents and orthogonality in the group algebra. We constructidempotent elements in F S n and study their orthogonality properties. Definition 1.21. Let T , . . . , T n ! be the tableaux of shape λ ⊢ n in lex order. For1 ≤ i, j ≤ n ! we define s ij ∈ S n by s ij T j = T i ; clearly s ji = s − ij and s ij s jk = s ik .We also define these group algebra elements (omitting λ if it is understood): D λi = H T i V T i = X h ∈ G H ( T i ) X v ∈ G V ( T i ) ǫ ( v ) hv. Proposition 1.22. If T is a tableau of shape λ ⊢ n then (1) D j = s ji D i s ij . Equivalently, (2) s ij D j = D i s ij . Proof. Using Proposition 1.20 and the definition of s ij , we obtain s ji D i s ij = s ji H T i V T i s − ji = (cid:2) s ji H T i s − ji (cid:3)(cid:2) s ji V T i s − ji (cid:3) = H s ji T i V s ji T i = H T j V T j = D j . This proves the first equation, and the second follows from s ji = s − ij . (cid:3) Proposition 1.23. If λ, µ ⊢ n with λ = µ , then D λi D µj = 0 for all tableaux T λi , T µj .Proof. We first assume Y λ ≺ Y µ . Proposition 1.15 shows that there exist twonumbers k, ℓ in the same row of T µ and the same column of T λ ; we now omit thesubscripts i, j . For the transposition t = ( k, ℓ ) we have t ∈ G V ( T λ ) and t ∈ G H ( T µ ).Using Lemma 1.19 we obtain D λ D µ = H T λ V T λ H T µ V T µ = H T λ V T λ t H T µ V T µ = H T λ ( V T λ t )( tH T µ ) V T µ = H T λ ( − V T λ )( H T µ ) V T µ = − H T λ V T λ H T µ V T µ = − D λ D µ . Hence D λ D µ = 0. On the other hand, if Y λ ≻ Y µ , then Proposition 1.20 impliesthat for any p ∈ S n we have H T λ pV T µ = H T λ (cid:0) pV T µ p − (cid:1) p = H T λ V pT µ p. Proposition 1.15 shows that there exist two numbers k, ℓ in the same row of T λ andthe same column of pT µ . Then t = ( k, ℓ ) ∈ G V ( pT µ ) ∩ G H ( T λ ) and so H T λ V pT µ p = H T λ t V pT µ p = ( H T λ t )( tV pT µ ) p = ( H T λ )( − V pT µ ) p = − H T λ V pT µ p. Hence H T λ V pT µ p = 0 and so H T λ pV T µ = 0, for all p ∈ S n . Therefore D λ D µ = H T λ V T λ H T µ V T µ = H T λ (cid:18) X p ∈ S n x p p (cid:19) V T µ = X p ∈ S n x p (cid:0) H T λ pV T µ (cid:1) = 0 , where x p ∈ F for all p ∈ S n . This completes the proof. (cid:3) Corollary 1.24. Let λ ⊢ n , and let T , . . . , T d λ be the standard tableaux in lexorder. If i > j then D i D j = 0 .Proof. We write H i , V i for H T i , V T i . By Proposition 1.17, there exist two numbers k, ℓ in the same column of T i and the same row of T j . Using the transposition t = ( k, ℓ ) and Lemma 1.19 we obtain D i D j = H i V i H j V j = H i V i t H j V j = H i ( V i t )( tH j ) V j = H i ( − V i )( H j ) V j = − D i D j . Therefore D i D j = 0. (cid:3) BREMNER, MADARIAGA, AND PERESI Proposition 1.25. (von Neumann’s Theorem) Let λ ⊢ n . For i = 1 , . . . , n ! wehave D i = c i D i where c i = n ! /f i , and f i is the dimension of the left ideal F S n D i .Proof. For scalars x p ∈ F which we will determine, we write D i = X p ∈ S n x p p. For any h ∈ G H ( T i ) and v ∈ G V ( T i ) we have hD i v = h (cid:18) X p ∈ S n x p p (cid:19) v = X p ∈ S n x p hpv,hD i v = ( hH i ) V i H i ( V i v ) = ǫ ( v ) H i V i H i V i = ǫ ( v ) D i . Therefore(3) X p ∈ S n x p hpv = ǫ ( v ) X p ∈ S n x p p. Each permutation in S n occurs once and only once on each side of this equation.First, consider the coefficient in D i of a permutation of the form hv . On the leftside of (3) take p = ι , on the right side take p = hv , and compare coefficients: x ι = ǫ ( v ) x hv . Hence x hv = ǫ ( v ) x ι . Second, consider the coefficient in D i of a permutation q notof the form hv . Lemma 1.16 implies that there are two numbers k, ℓ in the samerow of T i and the same column of qT i . For the transposition t = ( k, ℓ ), we have t ∈ G H ( T i ) and q − tq ∈ G V ( T i ). We can take h = t and v = q − tq in equation (3): X p ∈ S n x p tpq − tq = ǫ ( q − tq ) X p ∈ S n x p p. Setting p = q on both sides, we obtain x q tqq − tq = ǫ ( q − tq ) x q q, and this simplifies to x q q = − x q q , implying x q = 0. Combining the results of thetwo cases, we obtain D i = c i D i where c i = x ι .It remains to show that x ι = n ! /f i . We choose a basis for the left ideal F S n D i consisting of elements p D i , . . . , p f i D i where p , . . . , p f i ∈ S n , and extend thisto a basis of F S n . We regard D i as a linear operator on F S n , acting by rightmultiplication. The matrix representing D i with respect to our basis has the form (cid:20) x ι I f i ∗ (cid:21) , where ∗ indicates irrelevant entries. Hence trace( D i ) = x ι f i . On the other hand,since trace( q ) = 0 for q = ι , we havetrace( D i ) = trace (cid:18) X h,v ǫ ( v ) hv (cid:19) = X h,v ǫ ( v )trace( hv ) = trace (cid:0) I F S n (cid:1) = n ! . Now we have x ι f i = n !, so c i = x ι = n ! /f i . (cid:3) Definition 1.26. Let T λ , . . . , T λn ! be all the tableaux of shape λ ⊢ n . We define E λi = f i n ! D λi ( i = 1 , . . . , n !) . HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 9 Corollary 1.27. Every E λi is an idempotent: ( E λi ) = E λi . Two-sided ideals in the group algebra. The results in this subsectionlead us towards an explicit description of the isomorphism ψ in the Wedderburndecomposition (W). Definition 1.28. If T i , T j are tableaux of shape λ ⊢ n then we define ξ ij = ( ǫ ( v ) if s ji = vh for h ∈ G H ( T i ) and v ∈ G V ( T i )0 otherwise Lemma 1.29. If T i , T j are tableaux of shape λ ⊢ n then E i E j = ξ ij E i s ij .Proof. First, assume that s ji = vh for some h ∈ G H ( T i ), v ∈ G V ( T i ). Proposition1.22, equation (1) and von Neumann’s Theorem imply E i E j = E i ( s ji E i s ij ) = 1 c i H i ( V i v )( hH i ) V i s ij = 1 c i ǫ ( v ) H i V i H i V i s ij = ǫ ( v ) E i s ij = ǫ ( v ) E i s ij . Second, assume that s ji = vh for any h ∈ G H ( T i ), v ∈ G V ( T i ). Since T j = s ji T i ,Lemma 1.16 shows that there are numbers k, ℓ in the same column of T i and thesame row of T j . Using the transposition t = ( k, ℓ ) ∈ G V ( T i ) ∩ G H ( T j ) we obtain E i E j = 1 c i H i ( V i t )( tH j ) V j = − c i H i V i H j V j = − E i E j . Hence E i E j = 0. (cid:3) Remark 1.30. From now on we will work only with standard tableaux. Definition 1.31. Given a partition λ ⊢ n with standard tableaux T , . . . , T d λ inlex order, we write E λ for the d λ × d λ matrix with ( i, j ) entry ξ ij . Lemma 1.32. We have E λ = I λ + F λ where I λ is the identity matrix and F λ isa strictly upper triangular matrix. In particular, E λ is invertible.Proof. If i > j then Corollary 1.24 implies that E i E j = 0 and so ξ ij = 0. If i = j then s ii = ι and so Lemma 1.29 gives E i = ξ ii E i , hence ξ ii = 1. (cid:3) Proposition 1.33. If λ ⊢ n and T i , T j , T k , T ℓ are standard tableaux of shape λ then ( E i s ij )( E k s kℓ ) = ξ jk E i s iℓ . Proof. Using Proposition 1.22, equation (2), and Lemma 1.29, we obtain E i s ij E k s kℓ = s ij E j E k s kℓ = ξ jk s ij E j s jk s kℓ = ξ jk E i s ij s jk s kℓ = ξ jk E i s iℓ , as required. (cid:3) Remark 1.34. If we replace the scalar ξ jk in Proposition 1.33 by the Kroneckerdelta δ jk , and write E ij = E i s ij , then we obtain the matrix unit relations E ij E kℓ = δ jk E iℓ . In order to construct the isomorphism ψ , we need to modify the elements E i s ij to produce other elements which exactly satisfy the matrix unit relations. Definition 1.35. We write N λ for the the subspace spanned by the E λi s λij : N λ = span { E λi s λij | ≤ i, j ≤ d λ } ⊂ F S n . We write N for the sum of the subspaces N λ over all λ ⊢ n . Corollary 1.36. For each λ ⊢ n , the subspace N λ is a subalgebra of F S n . We fix a partition λ ⊢ n with standard tableaux T , . . . , T d λ in lex order. Let A = ( a ij ) be any d λ × d λ matrix over F , and consider the group algebra element(4) α λ ( A ) = d λ X i =1 d λ X j =1 a ij E i s ij . As usual, we write E ij for the d λ × d λ matrix with 1 in position ( i, j ) and 0 elsewhere. Lemma 1.37. For all partitions λ ⊢ n and all i, j, k, ℓ ∈ { , . . . , d λ } we have α λ ( E ij ) α λ ( E kℓ ) = α λ ( E ij E λ E kℓ ) . Proof. We have α λ ( E ij ) α λ ( E kℓ ) = E i s ij E k s kℓ = ξ jk E i s iℓ = α λ ( E ij E λ E kℓ ), usingProposition 1.33. (cid:3) Proposition 1.38. The set { E µi s µij | µ ⊢ n, ≤ i, j ≤ d µ } is linearly independent.Proof. A linear dependence relation among the E µi s µij can be written as X µ ⊢ n α µ ( A µ ) = 0 . We fix a partition λ , and obtain α λ (cid:0) E ii ( E λ ) − (cid:1)(cid:20) X µ ⊢ n α µ ( A µ ) (cid:21) α λ (cid:0) ( E λ ) − E jj (cid:1) = 0 . Using equation (2), Definition 1.26, and Proposition 1.23, we see that all termsvanish except for µ = λ : α λ (cid:0) E ii ( E λ ) − (cid:1) α λ ( A λ ) α λ (cid:0) ( E λ ) − E jj (cid:1) = 0 . Lemma 1.37 gives α λ ( E ii ( E λ ) − E λ A λ E λ ( E λ ) − E jj ) = 0, hence α λ ( E ii A λ E jj ) = 0and α λ ( a λij E ij ) = 0, and so a λij E i s ij = 0. Thus a λij = 0 for all λ and all i, j . (cid:3) Definition 1.39. Suppose that n has r distinct partitions λ , . . . , λ r in lex order.For i = 1 , . . . , r let d i = d λ i be the number of standard tableaux of shape λ i .Consider the direct sum of full matrix algebras M = r M i =1 M d i ( F ) . The linear map α : M → F S n is the direct sum of the α i = α λ i from equation (4): α ( A , . . . , A r ) = α ( A ) + · · · + α r ( A r ) . Corollary 1.40. The map α is injective. For every λ ⊢ n and ≤ i, j ≤ d λ , wehave dim N λ = d λ . The sum N of the N λ is direct, and hence dim N = P λ d λ .Proof. Injectivity of α is equivalent to the linear independence in Proposition 1.38.Since linear independence holds for each λ , the spanning set for N λ is also a basis.The sum of the N λ is direct by Proposition 1.23. (cid:3) Since N ⊆ F S n , it follows that P λ d λ ≤ n !, so to prove N = F S n , it remainsto show equality. Algorithms for insertion or deletion of a number to or from astandard tableau provide a bijection between S n and the set of ordered pairs ofstandard tableaux of the same shape. For details, see [27, § HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 11 Matrix units in the group algebra. We prove that the map ψ in (W) isan isomorphism by constructing elements of F S n corresponding to matrix units. Remark 1.41. The linear map α λ : M d λ ( F ) → F S n is not in general an algebrahomomorphism. However, we can easily obtain an algebra homomorphism from it. Definition 1.42. For all λ ⊢ n and 1 ≤ i, j ≤ d λ , we define the following elements: U λij = α λ (cid:0) E λij ( E λ ) − (cid:1) ∈ F S n . Proposition 1.43. For all λ, µ ⊢ n , ≤ i, j ≤ d λ , ≤ k, ℓ ≤ d µ we have U λij U µkℓ = δ λµ δ jk U λiℓ . Proof. If λ = µ then U ij U kℓ = α ( E ij E − ) α ( E kℓ E − ) = α ( E ij E − E E kℓ E − ) = α ( E ij E kℓ E − )= α ( δ jk E iℓ E − ) = δ jk α ( E iℓ E − ) = δ jk U iℓ . The factor δ λµ comes from the orthogonality of Proposition 1.23. (cid:3) Definition 1.44. We define the linear map ψ : M → F S n on matrix units as ψ ( E λij ) = U λij ( λ ⊢ n ; 1 ≤ i, j ≤ d λ ) . Theorem 1.45. The map ψ : M → F S n is an isomorphism of associative algebras.In particular, M d i ( F ) is isomorphic to N λ i .Proof. This is an immediate corollary of the preceding results. (cid:3) Remark 1.46. Since the direct sum M of full matrix algebras is clearly semisim-ple, and simplicity is preserved by isomorphism, it follows that F S n is semisimple,and moreover that it splits over F : the structure theory of semisimple associativealgebras implies that F S n is isomorphic to the direct sum of simple two-sided ideals,and that each simple ideal is isomorphic to the endomorphism algebra of a vectorspace over a division ring D over F . But our results show that D = F for every λ .Since the scalar factors d λ /n ! in Definition 1.26 are defined in characteristic > n ,we also obtain the semisimplicity of F S n in this case. Example 1.47. For n = 3 we take the permutations 123, 132, 213, 231, 312, 321in lex order – writing p as p (1) p (2) p (3) – as our basis of F S . The partitions λ = 3, µ = 21, ν = 111 have the following standard tableaux: T λ = 1 2 3 T µ = 1 23 T µ = 1 32 T ν = 123Thus d λ = 1, d µ = 2, d ν = 1 and hence we have the isomorphism ψ : M = F ⊕ M ( F ) ⊕ F −→ F S . As ordered basis of M we take the matrix units E λ , E µ , E µ , E µ , E µ , E ν . Wewill compute the corresponding elements U ρij of F S n . The groups of horizontal andvertical permutations are as follows: G H ( T λ ) = S G V ( T λ ) = { } G H ( T µ ) = { , } G V ( T µ ) = { , } G H ( T µ ) = { , } G V ( T µ ) = { , } G H ( T ν ) = { } G V ( T ν ) = S . The symmetric and alternating sums over these subgroups are as follows: H T λ = 123 + 132 + 213 + 231 + 312 + 321 V T λ = 123 H T µ = 123 + 213 V T µ = 123 − H T µ = 123 + 321 V T µ = 123 − H T ν = 123 V T ν = 123 − − 213 + 231 + 312 − D ρij are easily calculated; and scaling gives the idempotents: E λ = (123 + 132 + 213 + 231 + 312 + 321) E µ = (123 + 213 − − , E µ = (123 − − 231 + 321) E ν = (123 − − 213 + 231 + 312 − s µ = s µ = 132, and this is the only non-trivial case. Hence s = vh forany v ∈ G V ( T µ ), h ∈ G H ( T µ ) (see Lemma 1.29), and so every E ρ is the identitymatrix of size d ρ . Therefore every U ρij = α ρ ( E ij ) = E ρi s ρij , which gives the followingmatrix units in the group algebra: U λ = E λ U µ = E µ , U µ = E µ s = (132 + 231 − − U µ = E µ s = (132 − − 231 + 312) , U µ = E µ U ν = E ν These equations can be summarized by the matrix representing ψ with respect toour ordered bases of M and F S n , and then we obtain the matrix representing ψ − : ψ ∼ − 11 2 0 − − − 11 0 2 − − − − − − − ψ − ∼ − − 10 1 − − − − 11 0 − − − − − For any X ∈ F S , we have ψ − ( X ) = x E λ + x E µ + x E µ + x E µ + x E µ + x E ν = (cid:20) x , (cid:20) x x x x (cid:21) , x (cid:21) and therefore ψ − (123) = (cid:20) , (cid:20) (cid:21) , (cid:21) ψ − (132) = (cid:20) , (cid:20) (cid:21) , − (cid:21) ψ − (213) = (cid:20) , (cid:20) − − (cid:21) , − (cid:21) ψ − (231) = (cid:20) , (cid:20) − − (cid:21) , (cid:21) ψ − (312) = (cid:20) , (cid:20) − − (cid:21) , (cid:21) ψ − (321) = (cid:20) , (cid:20) − − (cid:21) , − (cid:21) These are the representation matrices for the irreducible representations of S . HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 13 Clifton’s theorem on representation matrices. Our next goal is to com-pute explicitly the algebra homomorphism φ : φ : F S n −→ M λ ⊢ n M d λ ( F )We fix λ ⊢ n throughout the following discussion, and consider all the tableaux T , . . . , T n ! of shape λ . Recall that for 1 ≤ i, j ≤ n ! we define s ij ∈ S n by theequation s ij T j = T i . If p ∈ S n then pT j = T r for some r , and so p = s rj . As before,we write E i for the idempotent corresponding to T i . Proposition 1.22 and Lemma1.29 show that E i E j = ξ ij E i s ij = ξ ij s ij E j . Therefore E i pE j = E i s rj E j = E i E r s rj = ξ ir s ir E r s rj = ξ ir s ir s rj E j = ξ ir s ij E j . We define ξ pij = ξ ir when p = s rj , so that for all i, j, p we have(5) E i pE j = ξ pij s ij E j . We now restrict to the d λ standard tableaux T , . . . , T d λ in lex order. Definition 1.48. For all p ∈ S n the Clifton matrix A λp is defined by( A λp ) ij = ξ pij (1 ≤ i, j ≤ d λ ) . The matrix previously denoted E λ is the Clifton matrix A λι for ι ∈ S n .Referring to the definition of ξ ij in Lemma 1.29, we see that A λp can be computedby the following steps, presented formally in Figure 3; see [14], [3], [12, Figure 1]: • Apply p to the standard tableau T j obtaining the (possibly non-standard)tableau pT j . • If there exist two numbers that appear together both in a column of T i andin a row of pT j , then ( A λp ) ij = 0. • Otherwise, there exists a vertical permutation q ∈ G V ( T i ) which takes thenumbers of T i into the rows they occupy in pT j . Then ( A λp ) ij = ǫ ( q ).Figure 3 attempts to find q , and returns 0 if no such permutation exists.Before proving Clifton’s theorem, it is worth quoting in its entirety the reviewin MathSciNet (MR0624907) by G. D. James of Clifton’s paper [14]: “From hisnatural representation of the symmetric groups, A. Young produced representationsknown as the orthogonal form and the seminormal form and gave a straightforwardmethod of calculating the matrices representing permutations. A disadvantage ofthese representations is that the matrix entries are not in general integers, and formany practical purposes, the natural representation is preferable. Most methods forworking out the matrices for the natural representation are messy, but this papergives an approach which is simple both to prove and to apply. Let T , T , . . . , T f be the standard tableaux. For each π ∈ S n , form the f × f matrix A π whose i, j entry is given by the following rule. If two numbers lie in the same row of πT j andin the same column of T i , then the i, j entry in A π is zero. Otherwise, the i, j entryequals the sign of the column permutation for T i which takes the numbers of T i to the correct rows they occupy in πT j . The matrix representing π in the naturalrepresentation is then A − I A π , where I is the identity permutation of S n .” Input: A partition λ = ( n , . . . , n ℓ ) ⊢ n and a permutation p ∈ S n .Output: The Clifton matrix A λp .For j = 1 , . . . , d λ do:(1) Compute pT j .(2) For i = 1 , . . . , d λ do:(a) Set e ← k ← β ← false .(b) While k ≤ n and not β do:(i) Set r i , c i ← row, column indices of k in T i .(ii) Set r j , c j ← row, column indices of k in pT j .(iii) If r i = r j then [ k is not in the correct row ] – if c i > n r j then set e ← β ← true [ required position does not exist ] – else if T i ( r j , c i ) < T i ( r i , c i ) then set e ← β ← true [ required position is already occupied ] – else set e ← − e , interchange T i ( r i , c i ) ↔ T i ( r j , c i )[ transpose k into the required position ](iv) Set k ← k + 1(c) Set ( A λp ) ij ← e Figure 3. Algorithm to compute the Clifton matrix A λp The Wedderburn decomposition of F S n shows that every permutation p ∈ S n isa sum of terms p λ ∈ F S n for λ ⊢ n , and each p λ is a linear combination of the U λij : p = X λ d λ X i =1 d λ X j =1 r λij ( p ) U λij Definition 1.49. We define R λ ( p ) to be the d λ × d λ matrix with ( i, j ) entry r λij ( p ).We call R λ ( p ) the representation matrix of p ∈ S n for λ ⊢ n . Lemma 1.50. We have U λii p U λjj = r λij ( p ) U λij . Proposition 1.51. (Clifton’s theorem) For all λ ⊢ n and p ∈ S n we have R λ ( p ) = ( A λι ) − A λp . Proof. We write E = A λι and denote the entries of E − by η ij . We have U λii p U λjj = α ( E ii E − ) p α ( E jj E − ) = d λ X k =1 η ik E i s ik ! p d λ X ℓ =1 η jℓ E j s jℓ ! = d λ X k =1 d λ X ℓ =1 η ik η jℓ E i s ik p E j s jℓ = d λ X k =1 d λ X ℓ =1 η ik η jℓ s ik E k p E j s jℓ (5) = d λ X k =1 d λ X ℓ =1 η ik η jℓ s ik ξ pkj s kj E j s jℓ = d λ X k =1 d λ X ℓ =1 η ik η jℓ ξ pkj s ik s kj E j s jℓ HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 15 = d λ X k =1 d λ X ℓ =1 η ik η jℓ ξ pkj s ik E k s kj s jℓ = d λ X k =1 d λ X ℓ =1 η ik η jℓ ξ pkj E i s ik s kj s jℓ = d λ X k =1 d λ X ℓ =1 η ik η jℓ ξ pkj E i s iℓ = d λ X k =1 η ik ξ pkj ! d λ X ℓ =1 η jℓ E i s iℓ ! = d λ X k =1 η ik ξ pkj ! U ij = ( A − ι A p ) ij U ij . Therefore r λij ( p ) = ( A − ι A p ) ij for all i, j and so R λ ( p ) = A − ι A p as required. (cid:3) Example 1.52. For n = 3 we have A λι = I d λ for all λ ⊢ 3, so R λp = A λp for all p ∈ S . Consider λ = 21 with d λ = 2, and p = 213. For i, j = 1 , T i and pT j , and the vertical permutation q (when it exists):( i, j ) = (1 , T = 1 23 pT = 2 13 q = ι ǫ ( q ) = 1( i, j ) = (1 , T = 1 23 pT = 2 31 q = 321 ǫ ( q ) = − i, j ) = (2 , T = 1 32 pT = 2 13 q does not exist( i, j ) = (2 , T = 1 32 pT = 2 31 q = 213 ǫ ( q ) = − ψ λ ) − (213) from Example 1.47: A λp = (cid:20) − − (cid:21) Example 1.53. Consider n = 5, the smallest n for which there exists λ ⊢ n suchthat A λι = I d λ . We list the standard tableaux for λ = 32 in lex order: T , . . . , T = 1 2 34 5 1 2 43 5 1 2 53 4 1 3 42 5 1 3 52 4Let p = ι and consider the ( i, j ) = (1 , 5) entry of E = A λι ; we have T i = T and pT j = T . The required vertical permutation is the transposition q = 15342interchanging 2 and 5, so ( A λι ) = − 1. Similar calculations show that A λι = I − E , ( A λι ) − = I + E . To illustrate the difference between the Clifton matrix A λp and the representationmatrix R λp = ( A λι ) − A λp , consider the 5-cycle p = 23451; in this case we obtain A λp = − − − − − R λp = − − − − − − . Computational methods for studying polynomial identities Let A be an algebra, not necessarily associative, which is finite dimensional overa field F . The multiplication in A is a bilinear map m : A × A → A denoted by( x, y ) xy . We write d for the dimension of A over F . If we choose an orderedbasis v , . . . , v d of the vector space A , then the multiplication in A can be expressedin terms of the structure constants c kij with respect to this basis: v i v j = d X k =1 c kij v k . A polynomial identity satisfied by A is an equation of the form I ≡ I isa nonassociative noncommutative polynomial (not necessarily multilinear or evenhomogeneous) which vanishes when arbitrary elements of A are substituted for thevariables in I . We use the symbol ≡ to indicate that the equation holds for allvalues of the variables. The polynomial identities satisfied by the algebra A do notdepend on the choice of basis.2.1. Historical background. We denote by T X ( A ) the set of polynomial identi-ties in the set of variables X satisfied by the algebra A . The set T X ( A ) is an idealof the free nonassociative algebra F { X } generated by X . Moreover, T X ( A ) is a T -ideal: f ( T X ( A )) ⊆ T X ( A ) for any endomorphism f : F { X } → F { X } . Problem 2.1. Specht, 1950 [39]. Given a class of algebras, determine whetherevery algebra in this class has a finite basis, in the sense that its T -ideal is generatedby a finite number of identities.Specht originally posed this problem for associative algebras over fields of char-acteristic zero. The complete solution was given by Kemer. Theorem 2.2. Kemer, 1987 [25] . Every associative algebra over a field of charac-teristic zero has a finite basis of identities. Similar results were obtained by Vais and Zelmanov [40] for finitely generatedJordan algebras (1989), and by Iltyakov [22, 23] for finitely generated alternativealgebras (1991) and Lie algebras (1992).If we consider the usual grading of F { X } by total degree, then we can study T X ( A ) n for each n ∈ N , the homogeneous component of degree n of the T -ideal.The nonzero elements of T X ( A ) n are the polynomial identities of degree n for A .An important problem is to find the smallest n for which T ( A ) n = 0; in this case,the nonzero elements of T X ( A ) n are called minimal identities for A . For the simplematrix algebras M n ( F ), the minimal identities were found by Amitsur and Levitzki. Theorem 2.3. Amitsur and Levitzki, 1950 [1] . The minimal degree of a polynomialidentity of M n ( F ) is n . Every multilinear polynomial identity of degree n for M n ( F ) is a scalar multiple of the standard polynomial: s n ( x , . . . , x n ) = X σ ∈ S n ǫ ( σ ) x σ (1) · · · x σ (2 n ) . Leron [28] proved (1973) that if char( F ) = 0 and n > n + 1 for M n ( F ) is a consequence of s n . In particular, theidentities of degree 7 for M ( F ) are consequences of s . Drensky and Kasparian[16] found (1983) all identities of degree 8 for M ( F ) when char( F ) = 0, and showed HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 17 that they are consequences of s ; see also Bondari [5, 6]. The T -ideal of identities T ( M ( F )) has been studied by many authors; see Razmyslov [33] for a survey. Thecomputational methods used to study polynomial identities of matrices have beendescribed by Benanti and co-authors [2]. Problem 2.4. Given an ordered basis and structure constants c kij for a finite-dimensional algebra A over F , determine the polynomial identities of degree ≤ n satisfied by A . In particular, find the minimal identities satisfied by A .Over a field of characteristic 0, every polynomial identity is equivalent to a setof multilinear identities; see Zhevlakov and co-authors [43]. Hence in character-istic 0, we may restrict our study to multilinear identities: equations of the form I ( x , . . . , x n ) ≡ I ( x , . . . , x n ) is a linear combination of monomials in whicheach of the n variables x , . . . , x n occurs exactly once. So each term of I ( x , . . . , x n )consists of a coefficient from F and a nonassociative monomial, which is a permuta-tion of the variables x , x , . . . , x n together with an association type (placement ofparentheses) which indicates the order in which the multiplications are performed.If there are t = t ( n ) association types in degree n , then I ( x , . . . , x n ) can be writtenas a sum of t summands I + I + · · · + I t , where the elements in each summand havethe same association type. Within each summand, the monomials differ only in thepermutation of the variables, and so each summand can be regarded as an elementof the group algebra F S n . We can therefore regard I ( x , . . . , x n ) as an element of F S n ⊕ · · · ⊕ F S n , the direct sum of t copies of F S n .This approach to polynomial identities was introduced independently in 1950by Malcev [29] and Specht [39]. In the 1970’s, Regev developed this theory fur-ther, with particular application to associative PI algebras; see for instance [34, 35].Around the same time, the computational implementation of this theory was initi-ated by Hentzel [18, 19]. (Two of the present authors learned about the applicationof this theory to polynomial identities through working with Hentzel.)2.2. Multilinear polynomial identities satisfied by an algebra. Consider acountable set X = { x , x , . . . } of variables. Definition 2.5. We construct the set M ( X ) of nonassociative monomials inthe variables X inductively as follows [43]: • X ⊂ M ( X ); • if x, y ∈ X and v, w ∈ M ( X ) \ X then xy, x ( v ) , ( v ) x, ( v )( w ) ∈ M ( X ).An association type is a placement of parentheses on a monomial of M ( X ).There is a natural total order on association types of degree n defined inductively,using the fact that every nonassociative monomial has a unique factorization x = yz .We fix a symbol ∗ and think of an association type as a monomial in which everyvariable equals ∗ . If x = yz and x ′ = y ′ z ′ are monomials of degree n , then x ≺ x ′ if and only if either y ≺ y ′ or y = y ′ and z ≺ z ′ . Example 2.6. For n = 3 we have two types: ( ∗∗ ) ∗ and ∗ ( ∗∗ ). For n = 4 we havefive types: (( ∗∗ ) ∗ ) ∗ , ( ∗ ( ∗∗ )) ∗ , ( ∗∗ )( ∗∗ ), ∗ (( ∗∗ ) ∗ ), ∗ ( ∗ ( ∗∗ )). Lemma 2.7. The number of distinct association types of degree n in F { X } is equalto the Catalan number (with the index shifted by 1): C n − = 1 n (cid:18) n − n − (cid:19) . The numbers C n grow very rapidly; here we present the first 12: n C n Example 2.8. If A is an associative algebra, then the placement of parenthesesdoes not affect the product, and so we only need to choose one association typein each degree as the normal form. If necessary, we choose the right-normed prod-uct x ( x ( · · · ( x n − x n ) · · · )), using the identity permutation of the variables. Butusually we can omit the parentheses and write simply x x · · · x n − x n . In an as-sociative algebra, any two multilinear monomials of degree n in n variables differonly by the permutation of the variables, and so a multilinear polynomial identityin degree n can be regarded as an element of the group algebra F S n . Example 2.9. If A is commutative (such as a Jordan algebra) or anticommutative(such a Lie algebra), then the association types are not independent. For example,we have ( ab ) c = ± c ( ab ). In these cases, the Wedderburn-Etherington numbers(oeis.org/A001190) enumerate the association types: 1, 1, 1, 2, 3, 6, 11, 23, 46, . . . Definition 2.10. If F is a field, then we write F { X } for the vector space over F with basis M ( X ). We define a multiplication on F { X } by extending bilinearly theproduct in M ( X ), and call this the free nonassociative algebra generated by X over F . Its elements are called nonassociative polynomials in the variables X . Definition 2.11. Let A be an algebra over F (not necessarily associative). A nonas-sociative polynomial f = f ( x , . . . , x n ) ∈ F { X } is called a polynomial identity of A if f ( a , . . . , a n ) = 0 for all a , . . . , a n ∈ A . We often write this more compactlyas f ≡ 0. If every variable x , . . . , x n appears exactly once in every monomial of f then f is called multilinear polynomial identity.We now explain the basic fill-and-reduce algorithm to find the multilinearpolynomial identities of degree n for an algebra A of dimension d < ∞ over a field F . We choose a basis for A and express elements of A as vectors in F d . In degree n there are t = t ( n ) association types and n ! permutations of the variables, for a totalof tn ! distinct monomials; we fix once and for all a total order on these monomials.A polynomial identity I ( x , . . . , x n ) is a linear combination of these tn ! monomials,with coefficients in F . Let E ( n ) be a matrix with tn ! columns and tn ! + d rows,consisting of a tn ! × tn ! upper block and a d × tn ! lower block. We generate n pseudorandom elements a , . . . , a n ∈ A . We evaluate the tn ! monomials by setting x i = a i ( i = 1 , . . . , n ) and obtain a sequence r j ( j = 1 , . . . , tn !) of elements of A . For each j we put the coefficient vector of r j into the j th column of the lowerblock. The d rows of the lower block consist of linear constraints on the coefficientsof the general multilinear polynomial identity I ( x , . . . , x n ). We compute the rowcanonical form RCF( E ( n )), so the lower block becomes zero. We repeat this processof generating pseudorandom elements of A , filling the lower block, and reducing thematrix until the rank of E ( n ) stabilizes. At this point, we write a for the nullity;the nullspace consists of the coefficient vectors of a canonical set of generatorsfor the multilinear polynomial identities satisfying the constraints imposed at eachstep, that is, the multilinear polynomial identities in degree n satisfied by A . Wecompute the canonical basis of the nullspace by setting the free variables equal tothe standard basis vectors and solving for the leading variables. We then put thesecanonical basis vectors into another matrix of size a × tn !, and compute its RCF, HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 19 which we denote by [All( n )]. We call the row space of this matrix All( n ); this is thevector space of all multilinear identities of degree n satisfied by A . This method isonly practical when the number tn ! of monomials is relatively small. Example 2.12. We find the polynomial identities of degree 4 for A = M ( F ), the4-dimensional associative algebra of 2 × F . We construct a 28 × E (4) and repeat the following process: • generate pseudorandom 2 × a , a , a , a over F ; • evaluate m ( j ) = a p j (1) a p j (2) a p j (3) a p j (4) for all p j ∈ S = { p , . . . , p } ; • for 1 ≤ j ≤ 24, store m ( j ) in the last 4 positions of column j of E (4): E (4) ,j ← m ( k )11 , E (4) ,j ← m ( k )12 , E (4) ,j ← m ( k )21 , E (4) ,j ← m ( k )22 ; • compute the row canonical form RCF( E (4)).The first 6 iterations produce ranks 4, 8, 12, 16, 20, 23 and the rank remains 23 forthe next 10 iterations. Hence the nullity is 1, and a basis for the nullspace consistsof the coefficient vector of the standard identity of degree 4:(6) s ( x , x , x , x ) = X p ∈ S ǫ ( p ) x p (1) x p (2) x p (3) x p (4) ≡ . This is the Amitsur-Levitzki identity from Theorem 2.3 in the case n = 2.2.3. Consequences of polynomial identities in higher degrees. When com-puting the multilinear polynomial identities satisfied by an algebra A , we often findthat many of the identities in degree n are consequences of known identities of lowerdegrees, so they do not provide any new information. We want the identities indegree n which cannot be expressed in terms of known identities of lower degrees. Definition 2.13. Let I ( x , . . . , x n ) be a multilinear nonassociative polynomialof degree n . There are n +2 consequences of this polynomial in degree n +1,namely n substitutions obtained by replacing x i by x i x n +1 ( i = 1 , . . . , n ) and twomultiplications of I by x n +1 (on the right and the left): I ( x x n +1 , . . . , x n ) , . . . I ( x , . . . , x i x n +1 , . . . , x n ) , . . . I ( x , . . . , x n x n +1 ) ,I ( x , . . . , x i , . . . , x n ) x n +1 , x n +1 I ( x , . . . , x i , . . . , x n ) . If I ≡ A , then so are its consequences. Lemma 2.14. Every multilinear polynomial of degree n +1 in the T -ideal gener-ated by I ( x , . . . , x n ) ∈ F { X } is a linear combination of permutations of the n +2 consequences in Definition 2.13.Proof. By definition, the T -ideal generated by I in F { X } is the ideal containing I which is invariant under all endomorphisms of F { X } . The n substitutions corre-spond to invariance under endomorphisms, and the two multiplications correspondto invariance under right and left multiplication. (cid:3) Example 2.15. The algebra O of octonions which we will study in detail lateris an example of an alternative algebra. Alternative algebras are defined by theleft and right alternative identities ( x, x, y ) ≡ x, y, y ) ≡ 0, where ( x, y, z ) =( xy ) z − x ( yz ) is the associator. Over a field of characteristic = 2, these two identitiesare equivalent to their linearized forms:( x, z, y ) + ( z, x, y ) ≡ , ( x, y, z ) + ( x, z, y ) ≡ . Each of these identities has five consequences in degree 4; for example, from theleft alternative identity we obtain( xw, z, y ) + ( z, xw, y ) ≡ , ( x, z, yw ) + ( z, x, yw ) ≡ , ( x, zw, y ) + ( zw, x, y ) ≡ , ( x, z, y ) w + ( z, x, y ) w ≡ , w ( x, z, y ) + w ( z, x, y ) ≡ . The vector space All( n ) of all multilinear polynomial identities of degree n sat-isfied by an algebra A is a subspace of the multilinear space of degree n in thefree nonassociative algebra F { X } where X = { x , . . . , x n } . Since All( n ) is invari-ant under permutations of the variables, we can regard All( n ) as a left S n -modulewith action given by permuting the subscripts of the variables: σ · f ( x , . . . , x n ) = f ( x σ (1) , . . . , x σ ( n ) ). We can also consider All( n ) as a submodule of Bin( n ), thedegree n component of the operad Bin generated by one nonassociative binaryoperation with no symmetry. The operad Bin is non-symmetric, but the operadgoverning the algebra A has the quotient Bin( n ) / All( n ) as its S n -module in degree n , and hence may be symmetric or non-symmetric, depending on the properties ofthe identities satisfied by A .For a given algebra A , the consequences in degree n of the identities of degrees < n generate a submodule Old( n ) ⊆ All( n ). We now explain the basic modulegenerators algorithm to find a canonical set of S n -module generators for Old( n ).We assume by induction that we have already determined a set of S n − -modulegenerators for All( n − n form aset O ( n ) of S n -module generators for Old( n ). We construct a ( tn ! + n !) × tn ! matrix C ( n ) consisting of a tn ! × tn ! upper block and a n ! × tn ! lower block (as before, t = t ( n ) is the number of association types in degree n ). Using the lex order onpermutations, we write σ i for the i th element of S n . We take an identity I ∈ O ( n )and for i = 1 , . . . , n ! we put the coefficient vector of σ · I into the i th row of thelower block. The n ! rows of the lower block then contain all the permutations of I ,and hence they span the S n -module generated by I . We compute RCF( C ( n )) sothe lower block becomes zero. We repeat this process for each I ∈ O ( n ). At theend, the nonzero rows of RCF( C ( n )) form a matrix [Old( n )] which contains thecoefficient vectors of a canonical set of S n -module generators for Old( n ).We compare the S n -modules Old( n ) and All( n ) to determine whether there existnew multilinear identities in degree n satisfied by A ; that is, identities which donot follow from those of degrees < n . To do this, we compare the reduced matrices[Old( n )] and [All( n )]; we denote their ranks by r old and r all . If r old = r all thenwe must have [Old( n )] = [All( n )]: every identity in degree n satisfied by A followsfrom identities of lower degrees. If r old = r all then since Old( n ) ⊆ All( n ) we musthave r old < r all , and the row space of [Old( n )] must be a subspace of the row spaceof [All( n )]. The difference r all − r old is the dimension of the S n -module of newidentities in degree n . Definition 2.16. The new identities satisfied by A in degree n are the nonzeroelements of the quotient module New( n ) = All( n ) / Old( n ). Definition 2.17. If X is a matrix in RCF, we write leading ( X ) for the set ofordered pairs ( i, j ) such that X has a leading 1 in row i and column j . We write jleading ( X ) = { j | ( i, j ) ∈ leading ( X ) } .We find S n -module generators for New( n ), by calculating the set difference jleading ([All( n )]) \ jleading ([Old( n )]) = (cid:8) j , . . . , j r (cid:9) ( r = r all − r old ) . HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 21 For s = 1 , . . . , r we define i s by ( i s , j s ) ∈ leading ([All( n )]). Lemma 2.18. Rows i , . . . , i r of [All( n )] are the coefficient vectors of the canonicalgenerators of New( n ) . Example 2.19. To illustrate these concepts, we extend the results of Example2.12 regarding 2 × E (5) has size 124 × E (5) has dimension 29;this is the S -module All(5), consisting of the coefficient vectors of all identities indegree 5 satisfied by 2 × s ( x x , x , x , x ) , s ( x , x x , x , x ) , s ( x , x , x x , x ) ,s ( x , x , x , x x ) , x s ( x , x , x , x ) , s ( x , x , x , x ) x . We construct a 240 × 120 zero matrix C (5) and do the following for each generator: • Set i ← • For each permutation p ∈ S do: – Set i ← i + 1. – For each term cm in the generator, where c = ± m = x q (1) · · · x q (5) ,let j be the index of pq in the lex-ordering on S , and set C (5) ij ← c . • Compute the row canonical form RCF( C (5)).After all 6 generators have been processed, the rank of C (5) is 24; its row space isthe S -module Old(5). Combining this result with that of the previous paragraph,we see that the quotient module New(5) has dimension 5.It remains to find generators for New(5). From RCF( E (5)) we extract a basisfor its nullspace, and sort these 29 vectors by increasing Euclidean norm (from 18to 74). Starting with RCF( C (5)) we apply the same module generators algorithmto these 29 vectors, and find that the first vector increases the rank from 24 to 29.Hence (the coset of) this single vector is a generator for New(5); this vector has 18(nonzero) terms, and all coefficients are ± E (5)) areall integers ( ± , ± × 120 integer matrix U with determinant ± U E (5) t is the Hermite normal form of the transpose of E (5). Thenthe bottom 29 rows of U form a lattice basis for the integer nullspace of E (5). Wesort these vectors by increasing Euclidean norm (from 16 to 34), and proceed as inthe previous paragraph. The first vector increases the rank from 24 to 29, and isthe coefficient vector of the linearization of the Hall identity:[ [ x , x ] ◦ [ x , x ] , x ] ≡ , where [ x, y ] = xy − yx is the Lie bracket and x ◦ y = xy + yx is the Jordan product.Drensky [15] has shown that s ≡ x, y ] , z ] ≡ T -ideal of identities satisfied by 2 × Representations of S n and multilinear identities in degree n . We canuse the representation theory of the symmetric group to break down these com-putations into smaller pieces, one for each irreducible representation of S n . Thissignificantly reduces the sizes of the matrices involved.Fix λ ⊢ n with irreducible representation of dimension d λ . Let E λij for i, j =1 , . . . , d λ be the d λ × d λ matrix units. It suffices to consider only the matrix unitsin the first row, in the following sense. Lemma 2.20. Let M λ be an irreducible submodule of type λ in the left regularrepresentation F S n . Then there exists a generator f ∈ M λ such that its matrixform φ λ ( f ) is in RCF and has rank 1 (the only nonzero row is the first).Proof. In the left regular representation, row i can be moved to row 1 by left-multiplying by the element of F S n which is the image under ψ of the elementarymatrix which transposes row 1 and row i . Recall that the matrix units in row i arelinear combinations of the elements E i s ij : combine Definitions 1.31, 1.42, 1.44 andequation (4). We can left-multiply by any p ∈ S n and obtain another element in thesame matrix algebra. In particular, if p = s i then using Proposition 1.22 we obtain s i E i s ij = E s i s ij = E s j . Thus left-multiplication by s i moves the matrix unitsin row i to row 1. The other rows are zero by the irreducibility assumption. (cid:3) Let t = t ( n ) be the number of association types in degree n . In the direct sumof t copies of the left regular representation, the λ -component is isomorphic to thedirect sum of t copies of the full matrix algebra M d λ ( F ). We construct a matrix M of size ( td λ + d ) × td λ , consisting of an upper block of size td λ × td λ and a lowerblock of size d × td λ . The multilinear associative polynomial U λ j of degree n is theimage under ψ of the matrix unit E λ j . Definition 2.21. For k = 1 , . . . , t we write [ U λ j ] k for the multilinear nonassociativepolynomial obtained by applying association type k to every term of U λ j .Given n pseudorandom elements of the algebra A , we can evaluate [ U λ j ] k usingthe structure constants of A to obtain another element of A . We do this for each k = 1 , . . . , t and each j = 1 , . . . , d λ to obtain a sequence of td λ elements of A ,which we regard as column vectors of dimension d . We store each of these columnvectors in the corresponding column of the lower block of M , and then computethe RCF( M ). We repeat this fill-and-reduce process until the rank of M stabilizes;at this point, the nullspace of M contains the coefficient vectors of the polynomialidentities satisfied by A in the component of ( F S n ) t corresponding to partition λ .We compute the canonical basis of the nullspace, and call its dimension a λ . Weput the basis vectors into another matrix of size a λ × td λ , and compute its RCF.This matrix, denoted allmat ( λ ), contains the canonical form of the polynomialidentities for A in partition λ .We need to compare allmat ( λ ) with the representation matrix for the conse-quences of known identities from lower degrees. We construct a matrix of size ℓd λ × td λ consisting of d λ × d λ blocks where ℓ is the number of consequences. Theblock in position ( i, j ) where i = 1 , . . . , ℓ and j = 1 , . . . , t is the representationmatrix for the terms of i th consequence in association type j . We compute theRCF of this matrix, and call its rank o λ . We denote the resulting o λ × td λ matrixof full rank by oldmat ( λ ); this contains the canonical form of all the consequencesin partition λ . HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 23 Since the row space of oldmat ( λ ) is a subspace of the row space of allmat ( λ )we have o λ ≤ a λ . Furthermore, oldmat ( λ ) = allmat ( λ ) if and only if o λ = a λ ; inthis case, there are no new identities for the algebra A in partition λ . We have jleading ( oldmat ( λ ) ) ⊆ jleading ( allmat ( λ ) ) . The rows of allmat ( λ ) whose leading 1s occur in the columns with indices in jleading ( allmat ( λ ) ) \ jleading ( oldmat ( λ ) ) , represent new identities for the algebra A in partition λ . (This is the representationtheoretic version of Lemma 2.18.)Consider one of the rows representing a new identity:[ c λ , . . . , c λ d λ , . . . , c λk , . . . , c λkd λ , . . . , c λt , . . . , c λtd λ ] (1 ≤ k ≤ t ) . As explained, we may assume that this is row 1 of the matrix, and so we can regardit as representing a linear combination of the elements [ U λ j ] k where 1 ≤ k ≤ t and1 ≤ j ≤ d λ , which gives an explicit form of the new identity: t X k =1 d λ X j =1 c λk,j [ U λ j ] k ≡ . In general, identities of this form have a very large number of terms, when fullyexpanded as elements of F S n , especially when n becomes large.2.5. The membership problem for T -ideals. A basic question about polyno-mial identities satisfied by an algebra is the following. Problem 2.22. Let f , . . . , f k and f be multilinear polynomial identities of de-gree n satisfied by an algebra A . Does f belong to the S n -module generated by f , . . . , f k ? Equivalently, is f a linear combination of permutations of f , . . . , f k ?Let φ λ : F S n → M d λ ( F ) be the projection onto the λ -component in the Wedder-burn decomposition (W). Let f = f + · · · + f t be the decomposition of f ∈ ( F S n ) t into terms corresponding to the t = t ( n ) association types. Definition 2.23. The representation matrix of f for λ equals: φ λ ( f ) = (cid:2) φ λ ( f ) | φ λ ( f ) | · · · | φ λ ( f t − ) | φ λ ( f t ) (cid:3) More generally, the representation matrix for a sequence of identities f , . . . , f k isobtained by stacking the matrices φ λ ( f ) , . . . , φ λ ( f k ): φ λ ( f , . . . , f k ) = φ λ ( f ) φ λ ( f )... φ λ ( f k ) = φ λ ( f ) φ λ ( f ) · · · φ λ ( f t − ) φ λ ( f t ) φ λ ( f ) φ λ ( f ) · · · φ λ ( f t − ) φ λ ( f t )... ... . . . ... ... φ λ ( f k ) φ λ ( f k ) · · · φ λ ( f kt − ) φ λ ( f kt ) Proposition 2.24. Let f , . . . , f k and f be multilinear polynomial identities ofdegree n . Then the following conditions are equivalent: • f belongs to the S n -module generated by f , . . . , f k • the matrices φ λ ( f , . . . , f k ) and φ λ ( f , . . . , f k , f ) have the same row space • the matrices φ λ ( f , . . . , f k ) and φ λ ( f , . . . , f k , f ) have the same RCF • the matrices φ λ ( f , . . . , f k ) and φ λ ( f , . . . , f k , f ) have the same rank Example 2.25. Every alternative algebra A satisfies the multilinear identity f ( x, y, z, t ) = ( xy, z, t ) + ( x, y, [ z, t ]) − x ( y, z, t ) − ( x, z, t ) y ≡ . To prove this we need to verify that f is a consequence of the alternative laws.Assuming char( F ) = 2, the alternative laws are equivalent to their linearizations:( x, y, z ) + ( y, x, z ) ≡ , ( x, y, z ) + ( x, z, y ) ≡ . The consequences of these identities in degree 4 are as follows; some follow fromothers using the alternative laws: f = ( xt, y, z ) + ( y, xt, z ) ≡ , f = ( xt, y, z ) + ( xt, z, y ) ≡ ,f = ( x, yt, z ) + ( yt, x, z ) ≡ , f = ( x, yt, z ) + ( x, z, yt ) ≡ ,f = ( x, y, zt ) + ( y, x, zt ) ≡ , f = ( x, y, zt ) + ( x, zt, y ) ≡ ,f = ( x, y, z ) t + ( y, x, z ) t ≡ , f = ( x, y, z ) t + ( x, z, y ) t ≡ ,f = t ( x, y, z ) + t ( y, x, z ) ≡ , f = t ( x, y, z ) + t ( x, z, y ) ≡ . In degree 4, there are t = 5 association types. For each λ ⊢ M λ = φ λ ( f , . . . , f ) , N λ = φ λ ( f , . . . , f , f ) , and compute their RCFs. For example, when λ = 22 we have d λ = 2 and so thematrix M λ has size 20 × 10 and N λ has size 22 × 10. We display N λ and its RCF,which coincides with the RCF of M λ : − − − − − − − − − − − − − − − − − − − − − − − 10 0 0 − − − − − − − 11 1 − − − − − − 10 0 0 0 0 0 − − 11 1 0 − − − − − − − − − − 10 0 1 0 0 0 0 0 − − 10 0 0 0 1 0 0 0 − − 10 0 0 0 0 0 1 0 − − HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 25 Further calculations show that for all λ ⊢ M λ and N λ are equal: λ d λ f ( x, y, z, t ) belongs to the S -module generated by the conse-quences in degree 4 of the linearized forms of the alternative laws.2.6. Bondari’s algorithm for finite-dimensional algebras. Bondari [5, 6] in-troduced an algorithm using the representation theory of S n which computes anindependent generating set for the multilinear identities (and central identities) ofthe full matrix algebra M k ( F ) with char F = 0 or char F = p > n where n is thedegree of the identities under consideration. He constructed all the multilinearidentities of degrees ≤ M ( F ), confirming existing results in the literature anddiscovering a new central identity in degree 8.Bondari’s algorithm can be used to find multilinear polynomial identities up toa certain degree (depending on computational limitations) for any algebra A over F of dimension d < ∞ . This algorithm involves evaluating matrix units in F S n usingthe structure constants of A with respect to a chosen basis. Definition 2.26. Fix λ ⊢ n and f = f + · · · + f t ∈ ( F S n ) t . The rank of thematrix φ λ ( f ) is called the rank of f for λ . If this rank is 1, then we say that f is irreducible for λ . (That is, the isotypic component of type λ in the submodulegenerated by f is irreducible.)Consider f ∈ ( F S n ) t and let r be the rank of the matrix RCF( φ λ ( f )). Each ofthe r nonzero rows g , . . . , g r generates an irreducible submodule of type λ , and theisotypic component of type λ is the direct sum of these r isomorphic submodules; inother words, r is the multiplicity of λ in the submodule generated by f . ExtendingLemma 2.20 to the case of t > g i can beregarded independently as an irreducible identity for λ in the first row of the matrix. Lemma 2.27. Every polynomial identity f ∈ ( F S n ) t is equivalent to a finite set ofidentities, each of which is irreducible for some λ ⊢ n .Proof. This is another way of saying that every finite dimensional S n -module over F is the direct sum of irreducible modules. (cid:3) Recall the images of the matrix units, U λ j = ψ ( E λ j ) ∈ F S n . The general element h ∈ F S n which is irreducible for λ ⊢ n has the form h = t X k =1 d λ X j =1 x k j [ U λ j ] k ( x k j ∈ F ) . Suppose that A has basis b , . . . , b d . We describe one iteration of Bondari’s algo-rithm. We choose arbitrary elements a , . . . , a n ∈ A and evaluate the [ U λ j ] k :[ U λ j ] k ( a , . . . , a n ) = d X i =1 c ikj b i . (This step can be very time-consuming, since the number of terms in the elements U λ j ∈ F S n is roughly n !.) Combining the last two equations we obtain h ( a , . . . , a n ) = d X i =1 t X k =1 d λ X j =1 c ikj x k j b i . If h is an identity for A then the coefficient of each b i must be 0 for all a , . . . , a n ∈ A : t X k =1 d λ X j =1 c ikj x k j = 0 (1 ≤ i ≤ d ) . This is a homogeneous linear system of d equations in the td λ coefficients x k j ofthe identity. We compute the RCF of the coefficient matrix, and find its rank.After s iterations, we have a linear system of sd equations. We repeat this processuntil the rank stabilizes. We then solve the system by computing the nullspace ofthe RCF. The nonzero vectors in the nullspace are (probably) coefficient vectors ofidentities satisfied by A . We need to check these identities by further computations.2.7. Rational and modular arithmetic. In general, we prefer to do all linearalgebra computations over the field Q of rational numbers. However, even if amatrix is very sparse and its entries are very small, computing its RCF can produceexponential increases in the entries. Even if enough computer memory is available tostore the intermediate results, the calculations can take far too long. It is thereforeoften convenient to use modular arithmetic, so that each entry uses a fixed smallamount of memory. This leads to the problem of rational reconstruction: recoveringcorrect results over Q or Z from known results over F p .Rational reconstruction is not well-defined: we try to compute an inverse for apartially-defined infinity-to-one map. It is only effective when we have a good the-oretical understanding of the expected results. For our computations, Remark 1.46explains why we may assume that the correct rational coefficients have n ! as theircommon denominator where n is the degree of the identities under consideration;see also [10, Lemma 8]. If we use a prime p > n ! then we can guess the common de-nominator b of the rational coefficients a/b from the distribution of the congruenceclasses modulo p : the modular coefficients are clustered near the congruence classesrepresenting a/b for 1 ≤ a ≤ b − 1. This allows us to recover the rational coefficients;we then multiply by the LCM of the denominators to get integer coefficients, andfinally divide by the GCD of the coefficients.Most of our computations require finding a basis of integer vectors for thenullspace of an integer matrix. In some cases, modular methods give good re-sults, meaning that the basis vectors have small Euclidean lengths. In other cases,we obtain much better results using the Hermite normal form (HNF) of an integermatrix together with the LLL algorithm for lattice basis reduction. If M is an s × t matrix over Z then computing the HNF of the transpose produces two matrices over Z : a t × s matrix H and a t × t matrix U with det( U ) = ± U M t = H .If rank( M ) = r then the bottom t − r rows of U form a lattice basis for the leftinteger nullspace of M t , which is the right integer nullspace of M . We then applythe LLL algorithm to obtain shorter basis vectors. For details, see [11, § a/b is nonzero in Q but zeroin F p : in lowest terms gcd( a, b ) = 1 and p | a . We assume that the probability of HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 27 this is 1 /p . We can make this entry 1 using rational arithmetic, but it will be 0 usingmodular arithmetic. If the algebra A has dimension d , then each iteration of thealgorithm produces another d linear equations in the coefficients of the polynomialidentity. So we expect to perform d operations of scalar multiplication of a rowduring the iteration. The chance that no error occurs is [1 − /p ] d . The chancethat an error occurs before the rank stabilizes, and remains for s iterations afterit stabilizes, is [1 − [1 − /p ] d ] s . For example, if we use p = 101 for an algebra ofdimension d = 8 and perform s = 10 iterations after the rank has stabilized, thenthe probability of incorrect results is ≈ . · − .We conclude this section with an important special case. Suppose that f ≡ A which has integralstructure constants with respect to a given basis. We multiply f by the LCM of thedenominators of its coefficients, obtaining a polynomial f ′ with integral coefficients;we then divide f ′ by the GCD of its coefficients, obtaining a polynomial f ′′ whosecoefficients are integers with no common factor. It is clear that f ′′ ≡ A , and that the reduction of f ′′ modulo p is nonzero for any primenumber p . Thus the existence of identities in characteristic 0 implies the existenceof identities in characteristic p for all p , and so non-existence in characteristic p fora single prime p implies non-existence in characteristic 0. Therefore we can verifynon-existence of identities over Q by computation over F p .2.8. Polynomial identities of Cayley-Dickson algebras. The most importantalternative algebra is the division algebra O of real octonions, which arises from theCayley-Dickson doubling process R ⊂ C ⊂ H ⊂ O ; see [43, § C ( α, β, γ ) depending on parameters α, β, γ ∈ F \ { } . Kleinfeld classifiedsimple alternative algebras in terms of Cayley-Dickson algebras. Theorem 2.28. Kleinfeld, 1953 [26]. A simple non-associative alternative algebrais a Cayley-Dickson algebra over its center. If F = R then C ( − , − , − 1) = O . If char F = 2 then it is possible to choose abasis 1 , e , . . . , e of C ( α, β, γ ) so that its multiplication table is Table 1.1 e e e e e e e e e e e e e e e e α e αe e αe − e − αe e e − e β − βe e e βe βe e e − αe βe − αβ e αe − βe − αβe e e − e − e − e γ − γe − γe − γe e e − αe − e − αe γe − αγ γe αγe e e e − βe βe γe − γe − βγ − βγe e e αe − βe αβe γe − αγe βγe αβγ Table 1. Multiplication table of the generalized octonions Problem 2.29. Find a basis for the T -ideal of polynomial identities of a Cayley-Dickson algebra C .Isaev [24] found a finite basis of T ( C ) when F is finite. Iltyakov [21] proved that T ( C ) is finitely generated when char F = 0 but did not give a set of generators.Racine [32] found the identities of degrees ≤ C when char F = 2 , , C over F such that every x ∈ C satisfies x − t ( x ) x + n ( x )1 = 0, wherethe trace t : C → F is a linear map and the norm n : C → F is a quadratic form. If x = a · P i =1 a i e i and x = a · − P i =1 a i e i are an element of C and its conjugate,then the trace and the norm of x are as follows: t ( x ) = x + x = 2 a,n ( x ) = xx = a − αa − βa + αβa − γa + αγa + βγa − αβγa . Theorem 2.30. Racine, 1985 [31]. Every quadratic algebra satisfies the identity V ( t ) − V ( t ) ◦ t ≡ , where x ◦ y = xy + yx , V x ( y ) = x ◦ y , and V = P σ ∈ S ǫ ( σ ) V x σ V y σ V z σ . It follows that every Cayley-Dickson algebra satisfies this identity. The identitiesof degree ≤ Theorem 2.31. Hentzel and Peresi, 1997 [20]. The identities of degree ≤ ofCayley-Dickson algebras are as follows, where either char F = 0 or char F = p > n ,and n is the degree of the identity: n ≤ no identities n = 3 ( x, x, y ) ≡ , ( x, y, y ) ≡ (alternative laws) n = 4 no identities n = 5 V ( t ) − V ( t ) ◦ t ≡ , [[ x, y ] ◦ [ z, t ] , w ] ≡ n = 6 h X σ ∈ S ǫ ( σ ) (cid:0) x ( y ( z ( tw ))) + 8 x (( y, z, t ) w ) − x, y, ( z, t, w )) (cid:1) , u i ≡ , where σ permutes x, y, z, t, w and ǫ is the sign.We give only the identities which are not consequences of those of lower degrees. In characteristic 0, Shestakov and Zhukavets [38] found a basis of three identities(one of degree 5 and two of degree 6) for the skew-symmetric identities of O . Incharacteristic = 2 , , 5, Shestakov [37] found a basis of identities for split Cayley-Dickson algebras C modulo the associator ideal of a free alternative algebra; that is,a basis for a homomorphic image T ′ ( C ) in the free associative algebra of the T -ideal T ( C ) of identities of C . Henry [17] found a basis for the Z -graded and Z -gradedidentities for Cayley-Dickson algebras (the latter case requires characteristic = 2).Bremner and Hentzel [8] studied identities for alternative algebras which are builtout of associators; in degree 7, they found two identities satisfied by the associatorin every alternative algebra, and five identities satisfied by the associator in O . HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 29 Multilinear identities for the octonions. We apply the computationaltechniques described in previous sections to the multilinear polynomial identitiessatisfied by the algebra O of octonions. We recover all the existing results in theliterature on identities in degree ≤ 6, and then show that there are no new identitiesin degree 7. As basis for O over the field F we take the symbols 1 , e , . . . , e . Thestructure constants depend on parameters α, β, γ ∈ F ; see Table 1. If F = R and α = β = γ = − Degree 3. Every multilinear identity in degree 3 satisfied by O follows from thelinearizations of the alternative laws; see [9, § 9, Example 1]. Degree 4. Every multilinear identity of degree 4 satisfied by O follows from theconsequences of the alternative laws; see [32]. We will verify this result usingour computational methods. The partitions λ ⊢ d λ = 1, 3, 2, 3, 1. The t = 5 association types are(( ∗∗ ) ∗ ) ∗ , ( ∗ ( ∗∗ )) ∗ , ( ∗∗ )( ∗∗ ) , ∗ (( ∗∗ ) ∗ ) , ∗ ( ∗ ( ∗∗ )) . We give details for λ = 22; the other cases are similar. The standard tableaux are:1 23 4 1 32 4The elements U λ , U λ ∈ Q S corresponding to the first row matrix units are U λ = ψ ( E λ ) = 1234 − − − − − − − − ,U λ = ψ ( E λ ) = 1324 − − − − − − − − . We create an 18 × 10 matrix consisting of 2 × × 10 upper blockand an 8 × 10 lower block. The columns correspond to the following elements of thedirect sum of t = 5 copies of F S , where the subscripts give the association types:[ U λ ] [ U λ ] [ U λ ] [ U λ ] [ U λ ] [ U λ ] [ U λ ] [ U λ ] [ U λ ] [ U λ ] Any identity for O of type λ can be expressed as a linear combination of theseelements. The fill-and-reduce algorithm converges after one iteration to this matrix: (cid:20) (cid:21) We find a basis for the nullspace and calculate its RCF, obtaining the matrix whoserows represent identities of type λ satisfied by O : allmat ( λ ) = − − 10 0 1 0 0 0 0 0 − − 10 0 0 0 1 0 0 0 − − 10 0 0 0 0 0 1 0 − − λ d λ r all r old r old+R1+R2 r old+R2+HP5 r old+R2 r old+HP5 Table 2. Multiplicities of irreducible modules in degree 5Using Clifton’s algorithm we obtain the matrix representing the 10 consequences indegree 4 of the alternative laws for partition λ ; this is M λ = φ λ ( f , . . . , f ) fromExample 2.25, whose RCF equals allmat ( λ ). Degree 5. Racine [32] found two new polynomial identities in degree 5 for O :(R1) [[ x, y ] , x ] ≡ , (R2) V ( t ) − V ( t ) ◦ t ≡ , where [ x, y ] = xy − yx , x ◦ y = xy + yx , and the square is with respect to themultiplication in O . For the definition of the operator V , see Theorem 2.30. Remark 2.32. The multilinear form of the identity (R2) can be written as x s +3 ( y, z, t ) − xs +3 ( y, z, t ) ◦ x ≡ , where s +3 ( x, y, z ) = s ( R ◦ ( x ) , R ◦ ( y ) , R ◦ ( z )) is an operator acting on the right, s isthe standard polynomial of degree 3, and R ◦ ( y ) the (right) multiplication operatorby y using ◦ : xR ◦ ( y ) = x ◦ y ; this follows the notation of [32].(R1) and (R2) are satisfied by O , but are not quite sufficient to generate New(5).Hentzel and Peresi [20] proved that [ v, w ] ◦ [ x, y ] is a central polynomial; that is,(HP5) [[ v, w ] ◦ [ x, y ] , z ] ≡ , is an identity of degree 5 for the algebra of octonions. The S -module New(5) isgenerated by (HP5) and (R2). Using our computational techniques, we obtained theresults summarized in Table 2. Column r all gives the multiplicity of the irreducible S -module [ λ ] in the module of all multilinear identities satisfied by O . Column r old gives the multiplicity of [ λ ] in the module of all consequences of the alternativelaws. Column r old+R1+R2 gives the multiplicity of [ λ ] in the module generated by theconsequences of the alternative laws and the two Racine identities (R1) and (R2).From this we see that (R1) and (R2) are sufficient in the first four representations,but in each of the last three representations, the multiplicities are one less thanrequired. Column r old+R2+HP5 gives the multiplicity of [ λ ] in the module generatedby the consequences of the alternative laws together with the identities (R2) and(HP5) these values are the same as r all for all λ , and the corresponding matrices areequal. The last two columns verify that, modulo the consequences of the alternativelaws, neither of the identities (R2) or (HP5) generates New(5) by itself, and thatthese two identities are independent (neither is implied by the other).We conclude this discussion by presenting explicit matrices to illustrate how wecan obtain new identities from the matrix units in the group algebra. For the lastpartition λ = 11111 with dimension d λ = 1, we obtain the matrices allmat( λ ) HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 31 allmat( λ ) = − − − − − − − − − − − − oldmat( λ ) = − − − − − − − − − − − − − − Figure 4. Matrices for new octonion identities ( λ = 11111) λ d λ r all r alt r old Table 3. Multiplicities of irreducible modules in degree 6and oldmat( λ ) displayed in Figure 4, with ranks of 11 and 10 respectively. Therow space of oldmat( λ ) is a subspace of the row space of allmat( λ ). Row 9 ofallmat( λ ) has a leading 1 in column 9, but oldmat( λ ) has no leading 1 in thiscolumn. Therefore row 9 of allmat( λ ) represents an identity satisfied by O whichis not a consequence of the alternative laws. In terms of matrix units, this row is E , − E , and therefore represents the following identity: X σ ∈ S ǫ ( σ ) h ( x σ (1) x σ (2) )( x σ (3) ( x σ (4) x σ (5) )) − x σ (1) (( x σ (2) x σ (3) )( x σ (4) x σ (5) )) i ≡ . Degree 6. Hentzel and Peresi [20] discovered a multilinear central polynomial ofdegree 5 for O , which produces the following polynomial identity where ( x, y, z ) =( xy ) z − x ( yz ) is the associator, and S permutes x, y, z, t, w :(HP6) h X σ ∈ S ǫ ( σ ) (cid:0) x ( y ( z ( tw ))) + 8 x (( y, z, t ) w ) − x, y, ( z, t, w )) (cid:1) , u i ≡ . Shestakov and Zhukavets [38] found a somewhat simpler central polynomial whichproduces the following polynomial identity:(SZ) h X σ ∈ S ǫ ( σ ) (cid:0) x, y ][ z, t ]) w − [[[[ x, y ] , z ] , t ] , w ] (cid:1) , u i ≡ . Using our computational techniques, we obtained the results in Table 3. Column r all gives the multiplicity of the irreducible S -module [ λ ] in the module of allmultilinear identities satisfied by O . Column r alt gives the multiplicity of [ λ ] inthe module of all consequences of the alternative laws. Column r old gives themultiplicity of [ λ ] in the module generated by the consequences of the alternativelaws and the identities (R2) and (HP5). From this we see that r old = r all except for λ = 111111 where the difference is 1; hence there is a new identity which alternatesin all 6 variables. We further checked that the multiplicities for the alternativelaws, (R2) and (HP5) together with either (HP6) or (SZ) are equal to r all for all λ ;hence either (HP6) or (SZ) can be taken as the new generator in degree 6.Our computations led us to the following new identity in degree 6, which involvesonly two of the 42 association types, alternates in all 6 variables, and does not havethe form [ f ( v, w, x, y, z ) , u ] ≡ f is a central polynomial:(7) X σ ∈ S ǫ ( σ ) (cid:16) x ( x (( x x )( x x ))) − x ( x ( x ( x ( x x )))) (cid:17) ≡ . We can use this identity instead of (HP6) or (SZ) as the new generator in degree 6. Degree 7. Our computations indicate that there are no new identities in degree 7. Theorem 2.33. Every multilinear polynomial identity of degree ≤ satisfied bythe octonion algebra O is implied by the consequences of the alternative laws, theidentities (R2) and (HP5) , and either (HP6) or (SZ) or identity (7) . We therefore conclude this paper with the following conjecture. Conjecture 2.34. The alternative laws together with the identities (R2) , (HP5) ,and either (HP6) or (SZ) or (7) , generate the T -ideal of polynomial identitiessatisfied by the octonion algebra O . Acknowledgements Murray Bremner was supported by a Discovery Grant from NSERC, the Nat-ural Sciences and Engineering Research Council of Canada. Sara Madariaga wassupported by a Postdoctoral Fellowship from PIMS, the Pacific Institute for theMathematical Sciences. Luiz Peresi thanks the Department of Mathematics andStatistics at the University of Saskatchewan for its hospitality and financial supportduring his visits in summer 2012 and spring 2014. HE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES 33 References [1] A. S. Amitsur, J. Levitzki : Minimal identities for algebras. Proc. Amer. Math. Soc. F. Benanti, J. Demmel, V. Drensky, P. Koev : Computational approach to polynomialidentities of matrices – a survey. 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