Course : Algebraic Combinatorics

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1 Course : Algebraic Combiatorics Lecture Notes # Addedum by Gregg Musier March 18th - 20th, 2009 The followig material ca be foud i a umber of sources, icludig Sectios , 7.7, , of Staley s Eumerative Combiatorics Volume 2. 1 Elemetary ad Homogeeous Symmetric Fuctios A polyomial i variables, P(x 1, x 2,...,x ) C[x 1, x 2,...,x ] is ow as a symmetric polyomial if for ay permutatio σ S, P(x σ(1), x σ(2),...,x σ() ) = P(x 1, x 2,...,x ). A importat family of symmetric polyomials is the family of elemetary symmetric fuctios. e = e (x 1, x 2,...,x ) := x i1 x i2 x i. 1 i 1 <i 2 < <i Notice that e 0 = 1, e (x 1, x 2,...,x ) = 0 if > ad the umber of terms i e (x 1, x 2,...,x ) is. If λ = [λ 1, λ 2,...,λ l ] is a partitio, e λ := e λ1 e λ2 e λl. (Fudametal Theorem of Symmetric Fuctios) Ay symmetric polyomial with coefficiets i C ca be writte as a C-liear combiatio of the e λ s. Furthermore, ay symmetric polyomial with coefficiets i Z ca be writte as a Z-liear combiatio of the e λ s. We will ot prove this theorem but will illustrate it for a few importat examples of symmetric fuctios. Let E(t) := =0 e t. The E(t) = i (1+x it). I particular, if we are worig with symmetric polyomials i variables, the i rages over {1, 2,..., } i this 1

2 product. Aother importat family of symmetric fuctios is family of homogeeous symmetric fuctios, defied as h = h (x 1, x 2,...,x ) := x i1 x i2 x i. 1 i 1 i 2 i Similarly we let h λ = h λ1 h λ2 h λl, h 0 = 1, h 1 = e 1, ad the umber of terms i h (x 1, x 2,...,x ) is, the umber of -elemet multisets of {1, 2,..., }. Let H(t) := =0 h t 1. The H(t) = i (1 x i. As a cosequece we get the t) followig result. Theorem. We have the idetity for all 1: ( 1) e i h i = 0. i=0 Proof. From the above, we see that H(t)E( t) = 1 so the covolutio { 1 if = 0 ( 1) i e i h i =. 0 if 1 i=0 As a corollary, we get that h = ( 1) i 1 e i h i. i=1 Thus by iductio, we get explicit expressios for h as a polyomial i terms of e 1 through e. Sice these idetities are true regardless of the umber of variables appearig i the polyomials, these are symmetric fuctio idetities rather tha simply idetities of polyomials. 2 Power symmetric fuctios We defie p = p (x 1, x 2,...,x ) := x 1 + x x,

3 the power symmetric fuctios, with p λ = p λ1 p λ2 p λl Theorem. These fuctios satisfy the Newto-Girard idetities for all 1: e = ( 1) i 1 e i p i h = h i p i. P(t) = p t. Notice that d H(t) = H (t) = h t 1, i=1 i=1 Proof. We prove the secod idetity, ivolvig the power symmetric fuctios ad the homogeeous symmetric fuctios. Let ad the logarithmic derivative H (t) H(t) = d log H(t) =1 =0 ( d ) = log (1 x i t) 1 i d = log(1 x i t) i ( d (x i t) ) j = j i j=1 ( ) = x j i t j 1 = j=1 i p t 1 = =1 P(t). t Thus P(t)H(t) = th (t) ad each coefficiet of t i the covolutio o the LHS, h i p i, eqauls the coefficiet of t 1 i H (t), amely h. i=1 The proof of the first idetity is aalogous. We leave it to the reader. As above, we ca use these idetities lie these to rewrite p s i terms of e λ s or h λ s, respectively, or vice-versa. First we itroduce some otatio. For i 1, let m i = m i (λ) copies of the umber i i λ. (Note that m i = 0 for i > λ.) z λ = i=1 im i (m i )!. Let ǫ λ = ( 1) m 2+m 4 +m

4 Lemma. If λ ad has l ozero parts, the ǫ λ = ( 1) l. I particular, ǫ λ is the sig of the permutatio that cotais m i (λ) i-cycles (for i 1). Proof. Left to the reader. Usig this otatio we obtai the followig result. Theorem. h = p λ λ z λ p λ e = ǫ λ z λ λ Proof. We saw i the last proof that d P(t) log H(t) =. t As a cosequece, ( ) p h t = exp t. =0 =1 The expoetial of a series, exp( =1 a t ) = exp(a(t)) equals the sum which ca be rewritte as the double sum i (a 1 t) r 1 (a 2 t 2 ) r 2 (a i t i ) r i i=0 uordered compostio r 1 +r 2 +r 3 + +r i =i each r j is a oegative iteger r 1, r 2,...,r i i! i=0 after expadig each term by the multiomial theorem. Sice the order of the compositio does ot matter, ad oly ozero parts cotribute to the summads, we ca thi of these r j s as the umber of j s i a partitio λ i, i.e. each such compositio gives rise to a λ so that r j = m j (λ). We the use the above otatio to rephrase this sum as m 1 m i (a i 1 am 2 2 a i ) t i exp(a(t)) =. m 1, m 2,...,m i i! i=0 λ i p p We leave as a exercise that the coefficiet of t i exp λ =1 t is λ z λ. A( t) i, i!

5 3 Moomial Symmetric Fuctios A eve simpler family of symmetric fuctios is the family of moomial symmetric fuctios. m λ = m λ (x 1, x 2,...,x ) := α x 1 1 α x [α 1,α 2,...,α ] is a rearragemet of [λ 1,λ 2,...,λ l,0,0,...,0] Observatio. e = m [1 ], p = m [], ad h = λ m λ. 2 2 x if > l, the umber of ozero parts i λ, ad we set m λ (x 1, x 2,...,x ) to be zero otherwise. (Note that whe we thi of m λ as a formal symmetric fuctio, i.e. i a ifiite umber of variables, this secod case ever occurs.) Remar. Note that ulie the e λ s, h λ s ad p λ s, m λ m λ1 m λ2 m λ. 4 Schur Fuctios We defie a fifth family of symmetric fuctios by usig determiats. Let Δ(x 1, x 2,...,x ) deote the determiat of the matrix x 1 x 2 x 3... x a δ = x 1 x 2 x 3... x x 1 x 2 x 3... x α Theorem. Δ(x 1, x 2,...,x ) = (x j x i ). 1 i<j Furthermore, Δ(x 1, x 2,...,x ) is the ozero polyomial with smallest degree ad the property that Δ(x σ(1), x σ(2),...,x σ() ) = sg(σ)δ(x 1, x 2,...,x ) for ay permutatio σ S. I particular, if σ is a traspositio that just switches x i ad x j, we get Δ(x 1, x 2,...,x ) o the RHS.

6 Such a polyomial is called a alteratig polyomial, ad it follows from above that all alteratig polyomials must be divisible by Δ(x 1, x 2,...,x ). We ca build other alteratig polyomials by taig the determiat of λ λ λ λ x 1 x 2 x 3... x λ 1 +1 λ 1 +1 λ 1 +1 λ x 1 x 2 x 3 x λ 2 +2 λ 2 +2 λ 2 +2 λ 2 +2 a λ+δ = x 1 x 2 x 3... x, λ 1 +( 1) λ 1 +( 1) λ 1 +( 1) λ 1 +( 1) x x x... x for ay partitio λ = [λ 1, λ 2,...,λ ] with at most parts, writte i wealy decreasig order. Cosequetly, the quotiet det(a λ+δ ) s λ = s λ (x 1, x 2,...,x ) = det(a δ ) is the quotiet of two alteratig polyomials, ad is i fact a symmetric polyomial (fuctio). We call these s λ s Schur fuctios. Remar. Note that lie the m λ s, s λ s λ1 s λ2 s λ. The Schur fuctios are very importat i the theory of represetatio theory of S ad GL. We will ot discuss such coectios further i the course, although there are may possible fial projects o this topic. There is a beautiful formula for writig the s λ s i terms of the h µ s (equivaletly the e µ s). The followig two formulas are ow as the Jacobi-Trudi Idetity. Theorem. If λ has l ozero parts, let JT l be the l-by-l matrix whose (i, j)th etry is h λi i+j, where we set h 0 = 1 ad h = 0 for < 0. The s λ = det JT l. Recall that λ T is the cojugate (or traspose) of λ. Let JT l (i, j)th etry is e λi i+j. The we also obtai be the matrix whose s λ T = det JT l. Example. h h h h 4 h 5 h 6 s 4,1 (x 1, x 2, x 3 ) = det h h h = det 1 h 1 h 2. h h h

7 (l) Proof. We let e j deote the jth elemetary symmetry fuctio o the alphabet {x 1, x 2,...,x l 1, x l+1,..., x }. ( )( 1 ) h i t i e (l) j ( t) j 1 = (1 x m t) 1 x i t i 0 j=0 i=1 m=1 m =l = = 1 + x l t + x l t x l t As a special applicatio, we tae the coefficiet of t α i o both sides ad obtai 1 (l) (l) α h i αi je ( 1) j = h αi +je ( 1) j j j = xl. j=0 j=1 This idetity implies the matrix equatio H α E = A α, α where we let the etries of A α be x i j s, the etries of H α be h αi +j s ad the etries of E be ( 1) i ) e (j i s. If we let α = [ 1, 2,..., 2, 1, 0] (resp. λ + [ 1, 2,..., 2, 1, 0]), the right-had-side gives precisely the etries of the matrix appearig i the deomiator (resp. umerator) of the Schur fuctio. It suffices to show that det E = det A [ 1, 2,...,2,1,0] = Δ(x 1, x 2,...,x ), ad thus we obtai det A λ+[ 1, 2,...,2,1,0] det H λ+[ 1, 2,...,2,1,0] =. det A [ 1, 2,...,2,1,0] The formula det E = det A [ 1, 2,...,2,1,0] follows from the fact that A [ 1, 2,...,2,1,0] = H [ 1, 2,...,2,1,0] E ad H [ 1, 2,...,2,1,0] is a upper triagular matrix with oes o the diagoal. We saw det A [ 1, 2,...,2,1,0] = Δ(x 1, x 2,...,x ) above. We close these otes with a alterative, more combiatorial defiitio, of Schur fuctios. We begi by geeralizig the defiitio of Stadard Youg Tableaux (SYT). Recall that a SYT of shape λ, λ, is a fillig of a Youg diagram of shape λ usig exactly the umbers {1, 2,..., } such that the umbers i each row icrease as we proceed to the right, ad the umbers i each colum icrease as we proceed dowwards.

8 A Semi-stadard Youg Tableaux (SSYT) of shape λ usig o umber smaller tha 1 or larger tha is a fillig of the Youg diagram so that the umbers i each row wealy icrease ad the umbers i each colum strictly decrease. s λ (x 1, x 2,...,x ) = x T. #i s appearig i T We defie the weight x T of a SSYT T to be the product i=1 x i. Theorem. SSYT T of shape λ usig o umber outside 1 i Proof. Omitted. The proof of this theorem alog with associated results or applicatios of SSYT is a possible fial project.

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