Simple proofs of some plethysm formulasWe present simple proofs of plethysm formulas for

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVII (1987)

S tanislaw B alcerzyk (Torun)

Simple proofs of some plethysm formulas

We present simple proofs of plethysm formulas for SpA 2(E) and S 2 A P(E), where E is a complex linear space of finite dimension, i.e., we give a reduction of those complex representations of a group G = GL(£), by explicit computation of generating vectors of heighest weight (we follow terminology of [2]). These formulas are due to Thrall [3] who proved them by the use of symmetric functions.

1. For each finite dimensional complex linear space E with a basis el5 eN and for each integer p ^ Г we have a map, natural with respect to linear maps of E

Cp,2: A 2p( E ) ^ E ® 2p —— >(A2(E))® p ->S p (A2(E)),

where v: E(g)E -> A 2(E) is a canonical epimorphism, Sp is a functor of pth component of a symmetric algebra functor and A denotes a comultiplication A' (E)-> A' (E)®A' (E) (induced by a map x i—>x(x)l + 1 ® x for x e E ) as well as its components A l+J(E)-* A l (E)<g>AJ(E) and compositions (as above). It is easy to check that for all x l5 ..., x 2pe E

(1) Zp .2 (*1 Л . . . л x 2p) = 2P - p l ' Z U u •••, Jp)Xj1 . . . xJp,

where J l f . . . , J p runs over all decompositions of {1,2, ..., 2p] in two element subsets, J l < ... < J p in lexicographic order and x ÿJi = x, л Xj for i

< j. Moreover, (J l5 ..., J p) is a sign of a corresponding permutation, where each J k is written in a natural order. It is easy to compute that

( 2 )

2 p

Ç p , l ( X 1 A . . . Л X 2 p ) = 2 p £ ( - l ) 4 p - i , 2 ( ^ 2 A . . . A X j л . . . Л X 2 p ) - X j Л X ; , 7 = 2

2 p + 1

(3) X ( - 1 У £p,2 (*1 A . . . Л Xj A . . . Л X2p+ x) ' X,- Л Xj = 0

7 = 1

for i' = l , . . . , 2p+ 1.

A map çPt2 transforms a vector of heighest weight ex л ... л e2p of a representation Л 2р(Е) of a group G = G L(£) onto non-zero vector w2p

= Çp,2 (*1 A ... A e2p) 6 Sp Л 2 (E).

T heorem 1. I f N ^ 2p then monomials w22 = w2Al . ..w 22s /o r all partitions A = (Alf A2, . . AJ о / p form a generating set of heighest weight vectors of SpA 2(E) thus

SPA 2(E) = © C(G)w2A^ © L 2 x (E),

A | - p Af - p

where L 2k = L 2k t2kg denotes a Schur functor corresponding to a Young diagram with columns of length 2A1, ..., 2AS.

P ro o f. For p = 1 the theorem is obvious. Let us assume that it holds for p. We use an epimorphic map of multiplication

mP, i : SPA 2(E)®A2( E) —>SP+1A 2(E). '

All monomials w2fl for p |- ( p + l ) are non-zero heighest weight vectors and have different weights. Thus Sp+lA 2(E) contains a G-submodule V

= © C{G)w2fl, and it is sufficient to prove that Immp l = К

рИ р+ i )

An inductive assumption and Pieri’s formula (see Remark 2) imply

f

(4) S2 A 2(E)®A2(E) * ® L 2X(E) ®A2(E) * ® L 2U+ei)(E)®@L2A+ej+Ek(E), where £,• denotes a sequence (0, ’..., 0, 1, 0, ...) with 1 in its ith place and the first .summation is over all A = (A1? ..., As) |- p and i ^ 1 such that Af_ x > A,, the second summation is over such A, j, к that l ^ j c / c ^ s + l and Ау_! > Aj ^ ... ^ Afc_! > Ak. To find heighest weight vectors corresponding to the decomposition (4) we define for q = 1, 2, ... natural maps

t]q: Л2я + ' (E) A£®<2«+ » v8>"®1 >(Л2(Е))8-®Е - S, Л2(£)®Е,

Л2ч + 2(Е) A 2(E) Л2(£)®Л2(£),

where As denotes a comultiplication in the symmetric algebra S A 2(E). Thus for x lt ..., x 2q + 2e E we have

2q + 1

riq{Xi л ... л Х2ч+1) = X ( - l ) a £„,2(*l A ... A xa A ... Л X2q+l)®*a,

a = 1

Cq (*1 A ... A X2q+2)

= 2 ( q + l ) X + л ... A Xa A ... A Xp A ... A X2q+2)®Xa Л Хдч

a </?

Vectors rjq(el л ... л e2q+l), Cq(^i л л ^iq + 2 ) are non-zero, then they are heighest weight vectors. Moreover,

™q,i °Cq = mqA o A s o £ q+lt2 = (q+ l)Zq+1,2-

Any vector in Sp Л 2(Е)®Л2(Е) admits a unique presentation in a form y = £ л е р; a non-zero term уаоц0®еЯо a ePo corresponding to a

tx </}

largest (with respect to a lexicographic order) pair (a0, fi0) is called a leading term of у and we write у = уаод0®<?яо а ePo + ...

(a) Let us fix A|—p and such i that Я,_1 > A, and let us denote т

= (Я1? ..., X,, ..., As), q = Я,-. The vector of heighest weight (5) mp_M <g) 1 (»V2t <g)£e (<?i A ... A f?2(J + 2 )

= 2(^+ 1) X ( - l ) a + /J_1 W2 t C9>2(C i a ... A £a A ... Л Сд A ... A É?2, + 2) <8>*?e A £д a </J

= 2(^ + 1) w2A® e2e+1 А в2ч + 2+ . . .

has weight corresponding to 2 (A+ £,•). Its image in 5Р+1Л 2(£) by mp>1 is W2t-We>1(Ce(ei л ... A é> 2 9+2)) - ( ? + l) w 2t- ^ +lf2(e1 A ... A é>2(J + 2)

= { q + l ) w 2U+ei) and this vector belongs to К

(b) Let us fix A (- p and j, к such that Aj_ ! > Xj ^ ^ Afc_ ! > Afc and let us denote £ = (Aj, ..., Àj, ..., Ak, ..., As), r — Xjy t = Xk. We define one more natural map

t]ry. A 2r+1 (E)®A2, + l (E) A^L-*Sf.A2(E)(g)E(S!S/ /12(£)® £ 1 —> Sr+i A Z(E)®A2(E).

The vector of heighest weight

(6) mp_r_ur+t® \ ( w 2e®qr4(e1 л ... л e2r+1 л ... л e2t+1))

= Z Z ( - i r ^ 2^ r , 2( « l A . . . A 4 A . . . A e 2r+1) a= 1 0=1

•£,t2(e 1 A ... А ёд A ... A e2f+1)®ea Л £д

= w2A(g)e2r+1 л e2t + 1+ ...

has weight corresponding to 2A+ £,- + £*. Its image in 5Р+1Л2(£) is zero,

because by (3) we get

£ ( - l )<x+pw2e^r,2(e1 a ... л ex a ... a e2r+ i)

«./»

2r+ 1

• ^ ,

Л . . . A f y Л . . . л e 2 t + l ) ' еа Л ^ = Z Л . . , A êp А . . . )

2r + 1

• Z Л ... л еа а ...)еа а ер = 0.

а= 1

Leading terms of vectors (5), (6) are linearly independent for all A, i, j, к then those vectors are linearly independent and consequently, they form a generating system of heighest weight vectors for all summands in (4). This

implies V = \ m m pl and the theorem follows.

R e m a rk 1. For each pair p, q of positive integers we define a natural map

£p<q: A pq(E) ~^E®pq A q(E))®p -» SpAq(E).

It is easy to write for £p>9 a formula analogous to (1) and to see that a map of transposition of two components in (A q(E))®p composed with v®poA is equal to ( — l)q2 v®poA. Thus Çp<q = 0 for odd q. Moreover, for even q, wp q

= CP>q(e 1 л . . . л €pq) Ф 0 is a vector of heighest weight in SpA q(E) and similarly as above the representation SpA q(E) contains a direct sum

© C(G)wqÀ ^ ® L qÀ(E), but is not equal to it for q > 2, p > 1.

2\- p

2. For each integer p ^ 1 and i = 0, 1, ..., p we have natural maps (pPti: А р+1(Е)®Ар~1(Е) - ^ ^ А Р{Е)®А'(Е)®АР~1(Е) -* A P(E)®AP(E).

It is easy to check that

Vi = (pPti{e ₁ л ... л ер+\ ® е г a ... a ep_,)

= ( _ l ) ( p - 0 i £ ( j ? I)eJ® e l a ... л ep-i л £?,,

where /, J runs over all decompositions of [1, 2, ..., p + i] such that \J\ = p,

|/| = i. Thus Vi Ф 0 is a vector of heighest weight and it generates a representation isomorphic to Lp+i>p_ ,•(£). By Pieri’s formula (see Remark 2) it follows A P(E)®AP{E) = ® C (G )v-.

It is clear that J, I term in the above sum is non-zero iff J = {1, ..., p — i}<jK with disjoint /, К contained in [p — i + 1, ..., p + i] and \K\ = i.

Consequently

Vi = ( - 1 ) (Р~1)1^ { К , I ) e t л ... л ep-i a eK® e x a ... a c p_,- a e,

and it is obvious that the transposition of components in A P(E)®AP(E) maps onto ( — l)'2 и,-. Thus the canonical map x p: A P(E)®AP(E) .—*■ S 2 ЛР(Е) maps V, onto non-zero vector iff i is even. We have proved

T

2. I f N ^ 2p then for j — 0, 1 [p/2] vectors x p(v2j) = 2 £ ( K , I ) e x л ... л ep_ 2j л е к -е1 л ... л ep_ 2J л

where К, I runs over all decompositions of {p — 2 j + l , . . . , p + 2j} into 2,}- element subsets, К < /, form a generating system of heighest weight vectors of S 2A p (E), then

IPl 2]

S 2 Ap(E)= © C(G)xp(v2j) * LPtP(E)@Lp+2'P_ 2®

j= о

where the last term is either L 2p(E) or

R e m a rk 2. For Pieri’s formula we can either invoke (5.17) in [1] (in terms of symmetric functions) or directly compute using a determinantal expression det (dx._.+ ) of a class of representation LX(E) in the representation ring of G, where dk corresponds to A k(E). This is particularly simple for A P(E)®AP(E) because

dp dp _{i= 0} I

dp—I — i ? dp+i +

dp-i Î LP+i,P-

i = 0

i ( E ) .

(

References

[1] I. G. M a c d o n a ld , Symmetric functions and Hall polynomials, Oxford University Press, 1979.

[2] M. A. N a im a r k , A. I. S tern , Theory o f group representations, Springer, Berlin 1982.

[3] R. M. T h r a ll, On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math. 64 (1942), 371-388.

INSTYTUT MATEMATYCZNY POLSKIF.I AKADFMII N A l’K MATHEMATICAL INSTITUTE POLISH ACADEMY OF SCIENCES