• Schur polynomials: \sum_[T in SSYT(N, m/n)] x^[content[T]]
• semistandard tableaux (increases in col, does not decrease in row, contains entires from {1..N})
• standard: contain each number in 1..n exactly once
• standard tableaux (increases in col, increases in row, contains entires from {1..N})

• power sums: p[k] = sum[i] x_i^k
• elementary symmetric polynomials: e[k](x1,...xn) = sum over all k-tuples of [x_1...xn] WITHOUT repetitioneg: e 2 = x1x2 + x2x3+ ... + x4x5. So this something like Sn(x^{subset of K})or Sn(x^(2^K)), or Σ[i1 < i2 < ... < in] x[i1] x[i2] .. x[in]
• complete symmetric polynomials: h k = sum over all k-tuples of [x1..xn ] WITH repetition eg: e[2](x1...x5) = x1^2 + x2^2 + ... x5^2 + , or multisets: Sn(x^{multiset of K}) or Σ[i1 ≤ i2 ≤ ... ≤ in] x[i1] x[i2] ... x[in].
• monomial symmetric polynomial m_ρ = Sn (x^ρ).

Examples for degree 2 in 3 variables:

• elementary: XY + YZ + XZ (variables of power 0/1, of degree 2.) (Alternatively, all subsets of size $2$ of $\{X, Y, Z\})$
• complete: X^2 + XY + YZ + XZ + Y^2 + Z^2 (all possible ways to get to degree 2). (Alternatively, all jjj
• monomial from 'X^2': 'X^2 + Y^2 + Z^2' (act with $S_3$ on the _variables_, generated by 'X^2').monomial from 'XY': 'XY + YZ + XZ' (act with $S_3$ on the _variables_, generated by 'XY').

• we're saying: take ALL possible ways with repetition. of course this will include all possibele ways to take "permuted".
• We are picking {1...N} with repetition $k$ times
• (a1, a2, .. an) | a1 + a2 + ... + an = k
• a[σ1], a[σ2]... a[σn] | aσ1 + aσ2 + ... + an = k [σ is a permutation]
• x1^5 x2^2 x3 + x2^5 x3^2 x4 + ... + x8^1 x7^5 x6^2
• mu = mu [1 ] >= mu [2 ] >= ... >= mu [n ].
• monominal symmetric polynomial indexed by mu: Informally whose ex-ponent vector can be rearranged to give μ.
• monomial symmetric polynomial: best way to make the monomial x^mu symmetric.

let i = σ-1[j]- f[1] = 5; f[2] = 3; f[3] = 1;x[1]^f[1] x[2]^f[2] x[3]^f[3] = x1^5 x2^3 x3σ 1 <-> 2<-><->x[σ[1]]^f[1] + x[σ[2]]^f[2] + x[σ[3]]^f[3]<-><->x[1]^f[σ-1[1]] + x[2]^f[σ-1[2]] + x[3]^f[σ-1[3]]<-><->g[1] = f[σ-1[1]] <->g[2] = f[σ-1[2]] <->g[3] = f[σ-1[3]] <-><->x[1]^g[1] x[2]^g[2] x[3]^g[3]

sort(f) = sort(g) = [5, 3, 1 ]

• Λn = all symmetric polynomials x1...xn. Vector space.
• The symmetric polynomials eα hα pα: let α be a sequence of [psotive integers (α1, ... αn) eα = Π_i eαi

x1, x1^2, x2, x2^2, ... x_k -> basis

• schur polynomials -> basis

• homogenous stuff.
• homogenous poly: all terms of the same degree. x1^2 + x2^2: homo. x1 + x2^2 NOT
• homogenous poly of degree k: form a vector space: (intuition: adding and scaling don't change degree)
• some symmetric polynomial:
• p(x) = x1x2^2 + x2x1^2 + x1x2 + x2x1 + x1 + x2
• p_1(x) = x1 + x2 ∈ Λ_2[1]
• p_2(x) = x1x2 + x2x1 ∈ Λ_2[2]
• p_3(x) = x1x2^3 + x2^2x1 ∈ Λ_2[3]
• p = p1 + p2 + p3

• What is a good example of a ring that is not a vector space over some field?

§ Basis for degree k with N variables:

• for each partition μ of k, we will have ONE basis element coresponding to μ, call it B(μ, k)
• Pick μ a partition of k
• B(μ, k) = \sum_{α: sort(α) = μ} x^α
• x^α = x1^α1 . x2^α2 . xN^αN

§ Example when k = 3, N = 4

• (a) 3 = 3 -> (3) -> (3, 0, 0, 0)
• (b) 3 = 2 + 1 -> (2, 1) -> (2, 1, 0, 0)
• (c) 3 = 1 + 1 + 1 -> (1, 1, 1) -> (1, 1, 1, 0)

• Consider all α such that sort(α) = (3, 0, 0, 0)
• '(3, 0, 0, 0)': X^3
• '(0, 3, 0, 0)': Y^3
• '(0, 0, 3, 0)': Z^3
• '(0, 0, 0, 3)': W^3
• B(μ, k, N) = \sum_{α: sort(α) = μ} x^α
• B((3, 0, 0, 0), 3, 4) = X^3 + Y^3 + Z^3 + W^3
• B((2, 1, 0, 0), 3, 4) = X^2Y + ...
• B((1, 1, 1, 0), 3, 4) = XYZ + YZW + ZWX + WXY

§ Partitions of a number $Par_N(k)$ and $Par(k)$

• $Par(k)$ are numbers $a_1 \geq a_2 \geq \dots$ such that $\sum_i a_i = k$
• $Par_N(k)$ are numbers $a_1 \geq a_2 \geq \dots a_N$ such that $\sum_i a_i = k$.That is, partitions of $k$ into exactly $N$ parts.

§ Symmetry of Schur polynomials

§ Kostka numbers

For each m/n shape and α sequence, define K(m/n, α) as the coefficient of x^α in s(m/n, x1, ..., xN).

Equivalenetly, K(m/n, α) is the number of semistandard tableaux of shape m/n and content α.

• for all skew shapes m/n and all alpha a, b such that sort(a) = sort(b) we have K(m/n, a) = K(m/n, b).

§ 10.41: Theorem: Lexicographic vs. Dominance Ordering.

Find better explanation (picture proof?)

If you have two non-decrasing functions μ and ν such that (integral_0^i μ < integral_0^i ν) for all i, then there exists a point $p$ such that μ(x < p) = ν(x < p) and μ(p) < ν(p)

§ Lemma 0 (Diaconis):

§ Definition of Raising Operation:

Restrict $R$ such that it only works for adjacent elements. May make life easier. Currently, we can move index $i$ to $j$ for $j < i$.

• ith row of t2:mu -> different columns of t1:lambda => lambda > mu
• ith row of t2:mu -> different columns of t1:lambda => mu < lambda

§ 1st row of mu

*1 *2 * * * * .. *p

§ 1st row of mu in lambda

  |  |  |  |..|..|  |..|      |  |  |  |..|..|  |>= p cols*3|  |*1|  |..|..|  |..|  |*4|  |  |..|..|  |..|  |  |  |*p|..|..|  |..|  |  |  |  |..|..|*2|..|

• #of cols of lambda >= # of cols of mu. l1 >= m1

§ 2nd row of mu

@1  @2 ... @q

§ 2nd row of mu in lambda

                         >= p cols*3|  |*1|  |..|..|  |..|..|@2|*4|  |  |@4|..|  |..|..|  |  |  |*p|..|..|  |..|..|  |  |  |  |..|..|*2|..|..|  |@1|  |  |..|..|@3|..|..|   |  |@q|  |..|..|  |..|..|       

pull up

                         >= p cols*3|*4|*1|*p|@4|..|*2|..|..|@2|@1|@q|  |..|..|@3|..|..|  |  |  |  |..|..|  |..|..|  |  |  |  |..|..|  |..|..|  |  |  |  |..|..|  |..|..|   |  |  |  |..|..|  |..|..|       

• the numbers fit in the first two rows.
• lambda have more space in the first two rows than mu.
• l1 + l2 >= m1 + mu.
• And so on. Pull up the values. We will have room in the first i rows.
• We can't just CHOP OFF the first row in lambda, because it has more cells, so we may lose some cells from mu. The only way to retain all the cells till step i is to argue about tableaux[:i].

§ Lemma 1 (diaconis):

Let $a, b$ be in the same row of $s$ and the same column of $t$. Consider $(a, b)$. Then $(a, b)e{s} = e{s}$, since ${s}$ is invariant under row permutation.

Also note that $[id, (a, b)]$ belong to $C_t$ as $a, b$ are in the same column of $t$. Consider the cosets of $[a, b]$ in $C_t$. This decomposes the full $C_t$ into .... This change the action into

sum_i (x_i id sgn(x_i id) + x_i (a, b) sgn (x_i (a,b))) = x_i (id - (a, b)) as sgn(id) = 1, sgn((a, b)) = -1

$M^\mu: module generated by all e_t$

§ Lemma 2(diaconis):

Recall that e[t]= Σ[pi ∈ Ct] sgn(π) e[pi{t}]

Let u in M_mu, that is, u is a linear combination of tabloids of shape mu. t is a mu-tableaux (not a TABLOID). Then for some scalar c,

Σ[pi ∈ Ct] sgn(pi) pi u = c e_t

A_t(u) = sgn(pi) pi (u) = c e_t

A_t(u) is the projector onto the e_t subspace.

Prove that A_t is idempotent, because 'A_t(e{t}) = +- e{t}' and 'A_t(e{s}) = 0' or 'A_t(e{s}) = +-e{s}'. So A_t projects onto at least the e[t] subspace. Can it project onto more? No, because we've seen that A_t(e{s}) = 0 or A_t(e{s}) = +-e{t}.

Next impose an inner product that makes the e{s} orthogonal. This shows that At is hermitian.

• S^λ = subspace spanned by e[t] of shape μ. Ie, the polytabloids.

§ Lemma 3

Vyn likes the statement as-is, since it talks about a fixed Slambda v/s varying U. I prefer to write it as Slambda in U or Slambda in Uperp.

consider A_t(U) for all t. It's either going to be et for SOME set of t's, or it's going to be identically zero. If it is identically zero, then we have Slambda perpendicular to U because Slambda is spanned by et. On the other hand, suppose A_x(U) is nonzero for some ex. Then, for all other y, we can find some permutation such that σ(x) = y. Since ex in U, and since U is invariant, we have that σ(ex) = eσ(x) = ey in U, which means that S is a subspace of U. So we show that one A_x being non-zero implies all the A_y are non-zero. Alternatively, I guess one can see that A_y = sgn(σ)σ A_x.