MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS. Contents

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1 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS Contents The Bibliography. Basic Notations. Classical Basis of Λ. Generating Functions and Identities 4 4. Frobenius transform and Hilbert series 4 5. Plethysm (λ-rings) 5 6. Macdonald symmetric functions 5 7. Macdonald Operators 6 8. Combinatorial aspects 7 9. q-analogs (see [Ber009] for more on this) 8 The Bibliography See also [a] [b] [b] [c]

2 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS. Basic Notations Partitions µ n iff µ = µ,, µ k ; µ µ k > 0; and n = µ j := µ. l(µ) = k. µ : l(c) i c j a(c) µ : leg arm n =,, : }{{} n times µ/λ : λ l µ (c) = l(c) := µ i+ (j + ); a µ (c) = a(c) := µ j+ (i + ); h µ (c) = h(c) := a(c) + l(c) + () := i d i d i! for µ = d n dn () n(µ) := i= c µ l(c) = a(c) = i (i,j) µ j and n(µ ) := c µ (i,j) µ Example clic here Wikipedia page on this Tableaux µ = µ,, µ k n iff µ N N; µ = { c c = (i, j), 0 j l(µ) ; 0 i µ j+ }; µ j = n. Tableau τ : µ {,,, n} Hook lenght formula : () f µ = n! c µ h(c) (4) Semi-Standard Tableau τ(a, j) < τ(b, j) a b and τ(i, c) < τ(i, d) c < d (f µ ) = n! µ n Example clic here Wikipedia page on this Standard Tableau f µ τ bijection, τ(a, j) < τ(b, j) a < b and τ(i, c) < τ(i, d) c < d Determinental formula : ( ( ) (5) f µ = n!det (µ i i + j)! i,j ). Classical Basis of Λ Λ = Λ Q : ring of symmetric functions ; x := {x, x, x, }. Basis are indexed by partitions, g = g(x). Monomial symetric functions (6) m µ := x µ i,,i k N distinct i x µ i x µ k i k Power sum symmetric functions : (7) p n := i Nx n i = m (n) and p µ := p µ p µk Complete homogeneous symmetric functions (8) h n := Elementary symmetric functions m λ and h µ := h µ h µk (9) e n := x i x in = m n and e µ := e µ e µk λ n where h 0 = e 0 = and e k = h k = 0 for all k < 0. i < <i n Schur symmetric functions (Jacobi-Trudi determinant formulas) (0) s µ = det ((h µi i+j) i,j ) ; s (n) = h n () s µ = det ((e µi i+j) i,j ) ; s n = m n = e n

3 Example clic here Wikipedia page on this MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS

4 4 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS Generating functions () E(t) := r 0 () H(t) := r 0. Generating Functions and Identities h r t r = i e r t r = i ( + x i t) ( x i t) = E( t) (4) n ( ) r e r h n r = 0 r=0 (5) P (t) := r 0 p r t r = d dt logh(t) = H (t) H(t) H(t) = ep (t) (6) nh n = (7) P ( t) := p r t r = d dt loge(t) = E (t) E(t) E(t) = e P ( t) r 0 (8) ne n = Wikipedia page on this Changing basis (5) (9) h n (x) = µ n The ω linear map ω : Λ Λ, p n ( ) n p n Scalar procduct } () p µ, p λ := δ µ,λ g, d = ω(g), ω(d) d, g Λ. n p r h n r r= n ( ) r p r e n r p µ (x) (7) (0) e n (x) = ( ) n l(µ) p µ (x), see also 9 and 0 µ n { ω (g(x)) = g(x), g Λ; ω(s µ ) = s µ ; Cauchy Kernel (4) Ω(xy) := i Changing basis r= () ω(h n ) = e n ; () ω(m µ ) = f µ, {f µ } is the forgotten base ( x i y j ) {f µ } and {g µ } dual basis iff d µ, g λ = δ µ,λ or Ω(xy) = µ d µ (x)g µ (y) (5) Ω(xy) = n h n (xy) = µ s µ (x)s µ (y) = µ h µ (x)m µ (y) = µ e µ (x)f µ (y) = µ p µ (x) p µ(y) Cyclic structure 4. Frobenius transform and Hilbert series Cyclic type λ(σ) = λ,, λ k iff σ = (σ,, σ λ ) (σ λ + +λ k +,, σ λ ) Class functions (6) R(S n ) := {χ : S n C χ(σ) = χ(τστ ), τ S n }; Characters σ = σ µ iff λ(σ) = µ χ V +χ W = χ V W and χ V χ W = χ V W Frobenius transform F (of an S n -module V ) (7) F(V ) = F(χ V ) := n! n! χ V (σ)p λ(σ) = χ V (σ λ )p λ σ S n λ n F(V W ) = F(V ) + F(W )

5 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS 5 Changing basis (8) F(χ µ ) = χ µ (σ λ )p λ = s µ, λ n where χ µ is an irreductible character. (9) F(χ Sn ) = p λ = h n, λ n where Sn is the trivial representation. (0) F(χ SignSn ) = λ n where Sign n χ SignSn (σ λ )p λ = µ n is the sign representation. Graded Frobenius characteristic () Frob q (V ) := n F(V n )q n, ( ) n l(µ) p µ = e n, Note 9 is equivalent to 9 and 0 is equivalent to 0. Bigraded Frobenius characteristic () Frob q,t (V ) := n F(V n,k )q n t k, where V = n V n is a graded S n -module. Hilbert series (poincaré series) () Hilb q (V ) := dim(v n )q n, n where V = n V n is a graded space. where, V = n,k V n,k is a bigraded S n -module. Bigraded Hilbert series (4) Hilb q,t (V ) := dim(v n,k )q n t k, n where, V = n,k V n,k is a bigraded space. plethysm is defined by : (5) p n [x + Y ] = p n [x] + p n [Y ], (6) p n [xy ] = p n [x]p n [Y ] Example 4 clic here 5. Plethysm (λ-rings) (7) p n [x] = x n therefor p n [p k (x)] = p nk (x), (8) p n [c] = c, if c is a constant, p n [qx] = q n p n (x) p n [tx] = t n p n (x) 6. Macdonald symmetric functions More scalar product... for original Macdonald polynomials...for combinatorial Macdonal polynomials l(µ) q µ i (9) p µ, p λ q,t = δ λ,µ i= t µ i l(µ) (40) p µ, p λ = ( ) µ l(µ) δ λ,µ ( q µ i )( t µ i ) (4) H µ, H λ = E µ (q, t)e µ(q, t)δ λ,µ, where E µ (q, t) = (q a(c) t l(c)+ ) and E µ(q, t) = c µ c µ(t l(c) q a(c)+ ) i= Cauchy formula for the H µ [ ] xy (4) e n = ( q)( t) µ n H µ (x; q, t)h µ (y; q, t) E µ (q, t)e µ(q, t)

6 6 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS Original Macdonald polynomials (Gram-Schmidt of the monomial basis in respect to,, q,t ) (4) P µ (x; q, t) = m µ + γ µ u γ (q, t)m γ Combinatorial Macdonald polynomials [ ] x (44) H µ (x; q, t) = P µ ; q, t (q a(c) t l(c)+ ) t c µ Example 5 clic here (45) H µ (x; q, ) = i H µi (x; q, ) (46) H µ (x; q, t) = H µ (x; t, q) [ ] x n [ ] x n (47) H n (x; q, ) = h n ( q i ) (48) H n (x; q, ) = e n ( q i ) q q i= i= (49) H µ (x; q, t) = Frob q,t (M µ ), where M µ = C{δx α δy β µ (x, y) α, β N n } is a Garcia-Haiman module and µ = det(x i k yj k ) k n. (i,j) µ Specialisation (50) H µ (x; 0, 0) = s n (5) H µ (x; 0, ) = h µ (5) H µ (x;, ) = s n (q, t)-kostka polynomials K λ,µ (q, t) (5) H µ (x; q, t) = λ µ K λ,µ (q, t)s λ (x), where K λ,µ (q, t) N[q, t]. (54) K λ,µ (q, t) = K λ,µ (t, q) (55) K λ,µ (q, t) = q n(µ ) t n(µ) K λ,µ(q, t ) (56) K λ,µ (t, q) = K λ,µ(q, t) 7. Macdonald Operators The operator (57) (H µ ) := q n(µ ) t n(µ) H µ (58) (Λ Z[q,t] ) Λ Z[q,t] and (Λ Z[q,t] ) Λ Z[q,t,/q,/t] (59) (e n ) = Frob q,t (DH n ), where DH = {f C[x, y] p h,k (δx, δy)f(x, y) = 0, h, k s.t. h + k > 0} is the diagonal harmonic space. (60) (e n ) t= = q area(γ) e ρ(γ), see figure. γ D n,n (6) (e n ), en = C n (q, t) Example 6 clic here The F operators (6) B µ := q i t j (6) F H µ (x; q, t) := F [B µ ]H µ (x; q, t) (64) F (Λ Z[q,t] ) Λ Z[q,t] (i,j) µ (65) F G = F G (66) F +G = F + G (67) cg = c G, for c Q ω and ω (68) ω (F (x; q, t)) := ω ( F ( x; q, t )) (69) ω (H µ (x; q, t)) = q n(µ ) t n(µ) H µ (x; q, t)

7 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS 7 (70) ω ω (H µ (x; q, t)) = (H µ (x; q, t)) (7) ω(h µ (x; q, t)) = q n(µ ) t n(µ) H µ (x; q, t ) (7) ed (e n ), F = ωf (e d ), s d, F Λ n µ n, ed (e n ), s µ = sµ (e d ), s d, The operator that multiplies by e The operator that differentiates by e (7) e : Λ d Q(q,t) Λ d+ Q(q,t) (74) δ e : Λ d Q(q,t) Λ d Q(q,t) H µ µ λ d λ,µ (q, t)h λ H µ λ µ c λ,µ (q, t)h λ d λ,µ (q, t) = q aµ(c) t lµ(c)+ t lµ(c) q aµ(c)+ c R λ,µ q a λ (c) t l λ (c)+ c C λ,µ, c t l λ (c) q a λ (c)+ λ,µ (q, t) = t lµ(c) q aµ(c)+ q aµ(c) t lµ(c)+ c R µ,λ t l λ (c) q a λ (c)+ c C µ,λ q a λ (c) t l λ (c)+ R λ,µ is the set of cells in the same row as λ/µ C λ,µ is the set of cells in the same column as λ/µ [ ] x It is proven that e is the adjoint of δ e in respect to,. ( q)( t) 8. Combinatorial aspects The Schur symmetric functions : (75) s µ := Example 7 clic here Pieri formula : Example 8 clic here (76) h n s µ = τ:µ xx τ, τ semi-standard and x τ = c µ θ n+ µ θ/µ ia a n-horizontal strip s θ. (77) e n s µ = x τ(c) The Kostka numbers K µ,λ (78) K µ,λ := #{ Semi-standard tableaux of shape µ fillings of λ} θ n+ µ θ/µ ia a n-vertical strip s θ. (79) s µ = λ n K µ,λ m λ, (80) h µ = λ n K λ,µ s λ, (8) e µ = λ n K λ,µ s λ. Example 9 clic here domino tabloid, d λ,µ (8) d λ,µ = #{ domino tableaux of shape λ and type µ} (8) e λ = µ n ( ) µ l(µ) d λ,µ h µ, (84) h λ = µ n( ) µ l(µ) d λ,µ e µ,

8 8 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS χ µ (λ) (85) χ µ (λ) := T ( ) ht(t ), summed over all border-strip tableaux, T, of shape µ and type λ, (86) ht(t ) = ht(t λi ) (87) s µ = λ n w λ,µ and v λ,µ χ µ (λ)p λ (88) w λ,µ = #{ Matrices of zeros and ones, with row sums λ and column sums µ } (89) v λ,µ = #{ Matrices of non negative integers, with row sums λ and column sums µ } (90) e λ = w λ,µ m µ, (9) h λ = v λ,µ m µ, µ n µ n Example 0 clic here 9. q-analogs (see [Ber009] for more on this) (9) [n] q := + q + q + q + + q n (9) [n]! q := [n] q [n ] q [] q [] q [ n [n]! q (94) := k] q [k]! q [n k]! q [ ] n (95) C n (q) := = [n] q[n ] q [n + ] q [n + ] q [n + ] q n q [n] q [n ] q [] q [] q [ ] [ ] n n + k (96) e k (, q, q,, q n ) = q k(k )/ (97) h k k (, q, q,, q n ) = k Example clic here q q

9 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS 9 Example : µ = , 7744 = 5, l(7744) = 5, µ = 5554 and λ = 4 : µ : µ : µ/λ : leg, l(c) = µ i+ (j + ) = cell, c = (, ) arm, a(c) = µ j+ (i + ) = 4 h 7744 (c) = a 7744 (c) + l 7744 (c) + = 8 hook, h 5554 (, ) = 5 = z 7744 = 7 4!!! = 9408; n(µ) = n(7744) = 9; n(µ ) = n(5554) = 57. Example : Tableau Semi-Standard Tableau Semi-Standard Tableau shape No order shape and filling (i.e. filled by {,, }) All standard tableaux of shape : shape and filling (i.e. filled by {,,,, }) f 5! 5 4 = = c = 5 ; 5! = h(c) 4 µ 5 (f µ ) = = 0

10 0 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS Example : m (x, y, z) = x y + x z + xy + xz + y z + yz h (x, y, z) = h h = (m + m )m = (x + y + z + xy + xz + yz)(x + y + z) e (x, y, z) = (xy + xz + yz)(x + y + z) p (x, y, z) = (x + y + z )(x + y + z) s (x, y, z) = x y + x z + xy + xz + y z + yz + xyz For n = 4 : h 4 (x, y, z) = m + m + m + m + m 4 h (x, y, z) = 4m + m + m + m + m 4 h (x, y, z) = 6m + 4m + m + m + m 4 h (x, y, z) = m + 7m + 4m + m + m 4 h (x, y, z) = 4m + m + 6m + 4m + m 4 s 4 (x, y, z) = h 4 s (x, y, z) = h h 4 s (x, y, z) = h h s (x, y, z) = h h h + h 4 s 4 (x, y, z) = e e + e + e e 4 s (x, y, z) = e e e + e 4 s (x, y, z) = e e s (x, y, z) = e e 4 s (x, y, z) = h h + h + h h 4 Example 4 : a) Let f = (q + t)p k, then : f [ 5qx t s (x, y, z) = e 4 ] [ = (q + 5qx ] t)pk t = (q + t) 5q n p t n k (x). b) p n [p (x)] = p n [x + x + ] = p n (x) = i N p n[x i ] = i N xn i ] c) p n [p k (x)] = p n [ i N xk i = i N p n[x k i ] = i N xkn i = p nk (x) p n [f(x)] = f[p n (x)] f Λ d) Let g = p (x) + p (x) and f = p (x) + p (x), then : and Therefor g[f(x)] f[g(x)]. g[f(x)] = (p + p )[f(x)] = p [p (x) + p (x)] + p [p (x) + p (x)] = p [p (x)] + p [p (x)] + (p [p (x) + p (x)]) = p [p (x)]p [p (x)] + p 6 (x) + (p (x) + p (x)) = p 6 (x) + p (x) + p (x) + p (x) + p (x) + p 6(x) f[g(x)] = (p + p )[g(x)] = (p [p (x) + p (x)]) + p [p (x) + p (x)] = p 6 (x) + p (x) + p (x) + p (x) + p 6(x)

11 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS Example 5 : H 4 (x, y, z) = q 6 s + ( q 5 + q 4 + q ) s + ( q 4 + q ) s + ( q + q + q ) s + s 4 H (x, y, z) = q s + ( q t + q + q ) s + ( q t + q ) s + ( q t + qt + ) s + ts 4 H (x, y, z) = q s + ( q t + qt + q ) s + ( q t + ) s + ( qt + qt + t ) s + t s 4 H (x, y, z) = qs + ( qt + qt + ) s + ( qt + t ) s + ( qt + t + t ) s + t s 4 H (x, y, z) = s + ( t + t + t ) s + ( t 4 + t ) s + ( t 5 + t 4 + t ) s + t 6 s 4 Figure. Dyck path, γ D 5 with riser ρ(γ) = Figure. Dyck path of area 4 C n := #D n Example 6 : t= (e ) = q e + q e + qe + qe + e Example 7 : s (x, y, z) = x y z y z z z z x + x x + x y + x z + y y + y z + x y + x y z = x y + x z + xy + xz + y z + yz + xyz Example 8 : h s = = = s 5 + s 4 + s 4 + s Example 9 : K,5 = 0, K,4 = 0, K, = 0, K, = 0, K, =, K, =, K, = 5 s = m + m + 5m

12 MATHEMAGICAL FORMULAS FOR SYMMETRIC FUNCTIONS K 5, =, K 4, =, K, =, K, =, K, =, K, = 0, K, = 0 h = s 5 + s 4 + s + s + s e = s + s + s + s + s Example 0 : w, = #{} = 0, w, = # {[ ]} 0 =, w, = # 0 0 0, , = e = m + m v, = # {[ ]} {[ ] 0 =, v 0, = #, 0 h = m + m + m, [ ]} 0 =, v 0, = w, = Example : [5] q = q 4 + q + q + q + [4]! q = (q + q + q + )(q + q + )(q + )() [ 8 4] q = [8] q[7] q [6] q [5] q [4] q [] q [] q [] q [4] q [] q [] q [] q [4] q [] q [] q [] q C 4 (q) : = [ 8 = [5] q 4] [8] q[7] q [6] q q [4] q [] q [] q [] q = + q + q + q 4 + q 5 + q 6 + q 7 + q 8 + q 9 + q 0 + q