## Symmetric functions

Last update: 04 March 2012

## Symmetric functions

The symmetric group ${S}_{n}$ acts on the vector space $ℤn= ℤ-span{x1,...,xn} by wxi=xw(i),$ for This action induces an action of ${S}_{n}$ on the polynomial ring $ℤ\left[{X}_{n}\right]=ℤ\left[{x}_{1},...,{x}_{n}\right]$ by ring automorphisms. For a seq uence $\gamma =\left({\gamma }_{1},...,{\gamma }_{n}\right)$ of non-negative integers let The ring of symmetric functions is Define the orbit sums, or monomial symmetric functions, by where ${S}_{n}\lambda$ is the orbit of $\lambda$ under the action of ${S}_{n}.$ Let so that

### Partitions

A partition is a collection $\mu$ of boxes in a corner where the convention is that gravity goes up and to the left. As for matrices, the rows and columns of $\mu$ are indexed from top to bottom and left to right, respectively. Then $\mu$ is determined by (and identified with) the sequence $\mu =\left({\mu }_{1},...,{\mu }_{l}\right)$ of positive integers such that ${\mu }_{1}\ge {\mu }_{2}\ge \cdots \ge {\mu }_{l}>0,$ where $l=l\left(\mu \right).$ For example, $(5,5,3,3,1,1)= .$ A partition of k is a partition $\lambda$ with $k$ boxes. Write $\lambda &rline;k$ if $\lambda$ is a partition of $k.$ Make the convention that ${\lambda }_{i}=0$ if $i>l\left(\lambda \right).$ The dominance order is the partial order on the set of partitions of $k,$ given by PUT THE PICTURE OF THE HASSE DIAGRAM FOR $k=6$ HERE.

### Tableaux

Let $\lambda$ be a partition and let $\mu =\left({\mu }_{1},...,{\mu }_{n}\right)\in {ℤ}_{\ge 0}^{n}$ be a sequence of nonnegative integers. A column strict tableau of shape $\lambda$ and weight $\mu$ is a filling of the boxes of $\lambda$ with ${\mu }_{1}$ 1s, ${\mu }_{2}$ 2s, ... , ${\mu }_{n}$ $ns$, such that

1. the rows are weakly increasing from left to right,
2. the columns are strictly increasing from top to bottom.
If $p$ is a column strict tableau write $\mathrm{shp}\left(p\right)$ and $\mathrm{wt}\left(p\right)$ for the shape and the weight of $p$ so that For example, $p= 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 4 3 3 3 4 4 4 5 4 5 5 6 6 7 7 has shp(p) = (9,7,7,4,2,1,0) and wt(p) = (7,6,5,5,3,2,2).$ For a partition $\lambda$ and a sequence $\mu =\left({\mu }_{1},...{\mu }_{n}\right)\in {ℤ}_{\ge 0}$ of nonnegative integers write

### Elementary symmetric functions

Define symmetric functions ${e}_{r},$ $0\le r\le n,$ via the generating function $∏i=1n (1-xiz) = ∑r=0n (-1)rerzr.$ Then ${e}_{0}=1$ and, for $0\le r\le n,$ $er=m(1r) = ∑ 1≤i1 where the last sum is over all column strict tableaux $p$ of shape $\left({1}^{r}\right).$

If $f\left(t\right)$ is a polynomial in $t$ with roots ${\gamma }_{1},...,{\gamma }_{n}$ then If $A$ is an $n×n$ matrix with entries in $𝔽$ with eigenvalues ${\gamma }_{1},...,{\gamma }_{n}$ then the trace of the action of $A$ on the ${r}^{\mathrm{th}}$ exterior power of the vector space ${𝔽}^{n}$ is and the characteristic polynomials of $A$ is $chart(A) = ∑r=0n (-1)n-r en-r (γ1,...,γn)tr.$

Let $\gamma =\left({\gamma }_{1},...,{\gamma }_{n}\right)$ be a partition. Then $eλ' = ∑ μ≤λ aλ'μmμ,$ where ${a}_{\lambda \text{'}\mu }$ is the number of matrices with entries from $\left\{0,1\right\}$ with row sums $\lambda \text{'}$ and column sums $\mu .$ Furthermore, ${a}_{\lambda \text{'}\lambda }=1$ and ${a}_{\lambda \text{'}\mu }=0$ unless $\mu \le \lambda .$

 Proof. If $A$ is an $l×n$ matrix with entries from $\left\{0,1\right\}$ let $xA = ∏i=1n (xi)aij$ and define so that $rs\left(A\right)$ and $cs\left(A\right)$ are the sequences of row sums and column sums of $A$, respectively. If $\lambda \text{'}=\left({\lambda }_{1}\text{'},...,{\lambda }_{l}\text{'}\right)$ then $eλ' = ∏j=1l eλj' = ∑ rs(A)=λ' xA = ∑ γ∈ℤ≥0n ∑ rs(A)=λ' cs(A)=γ xγ = ∑ μ aλ'μ mμ.$ Since there is a unique matrix $A$ with $rs\left(A\right)=\lambda \text{'}$ and If $A$ is a $0,1$ matrix with $rs\left(A\right)=\lambda \text{'}$ and $cs\left(A\right)=mu$ then ${\mu }_{1}+\cdots +{\mu }_{i}\le {\lambda }_{1}+\cdots +{\lambda }_{i}$ since there are at most ${\lambda }_{1}+\cdots +{\lambda }_{i}$ nonzero entries in the first $i$ columns of $A.$ Thus ${a}_{\lambda \text{'}\mu }=0$ unless $\mu \le \lambda .$ $\square$

1. The set is a basis of $ℤ{\left[{X}_{n}\right]}^{{S}_{n}}.$
2. $ℤ{\left[{X}_{n}\right]}^{{S}_{n}}=ℤ\left[{e}_{1},...,{e}_{n}\right].$

### Complete symmetric functions

Define symmetric functions ${h}_{r}$, $r\in {ℤ}_{\ge 0},$ via the generating function $∏i=1n 1 1-xiz = ∑ r∈ℤ≥0 hrzr.$ Then ${h}_{0}=1$ and, for $r\in {ℤ}_{>0},$ $hr = ∑ λ⊢r mλ = ∑ 1≤i1≤i2≤⋯≤ir≤n xi1xi2⋯xir = ∑ sh(p)=(r) xwt(p),$ where the last sum is over all column strict tableaux $p$ of shape $\left(r\right).$

There is an involutive automorphism $\omega$ of $ℤ{\left[{X}_{n}\right]}^{{S}_{n}}$ defined by $ω: ℤ[Xn]Sn → ℤ[Xn]Sn ek ↦ hk$

 Proof. Comparing coefficients of ${z}^{k}$ on each side of $1= ∏i=1n (1-xiz) ∏i=1n 1 1-xiz yields 0= ∑r=1k (-1)r erhn-r.$ $\square$

1. The set is a basis of $ℤ{\left[{X}_{n}\right]}^{{S}_{n}}.$
2. $ℤ{\left[{X}_{n}\right]}^{{S}_{n}}=ℤ\left[{h}_{1},...,{h}_{n}\right].$

The monomials in form a basis of $ℤ\left[{x}_{1},...,{x}_{n}\right]$ as a ${ℤ\left[{x}_{1},...,{x}_{n}\right]}^{{S}_{n}}$ module.

 Proof. Let $I=⟨{e}_{1},...,{e}_{n}⟩$ be the ideal in $ℤ\left[{x}_{1},...,{x}_{n}\right]$ generated by ${e}_{1},...,{e}_{n}.$ Since and so $∑r=0n-i (-1)r er (xi+1,...,xn)tr = ∑ l≥0 hl(x1,...,xi)tl modI.$ Comparing coefficients of ${t}^{n-i+1}$ on each side gives that, for all $1\le i\le n,$ $0=hn-i+1(x1,...,xi) = ∑r=0n-i+1 xn-i+1-r hr(x1,...,xi-1) modI,$ and thus $xin-i+1 = - ∑r=1n-i+1 xn-i+1-r hr(x1,...,xi-1) modI. (Sf 1.11)$ This identity shows (by induction on $i$) that ${x}_{i}^{n-i+1}$ can be rewritten, $\mathrm{mod}I,$ as a linear combination of monomials in ${x}_{1},...,{x}_{i}$ with the exponent of ${x}_{i}$ being $\le n-i.$ In particular, and it follows that any polynomial can be written, $\mathrm{mod}I,$ as a linear combination of monomials $x1ε1x2ε2⋯xnεn with 0≤εi≤n-i. (Sf 1.12)$ If ${S}^{k}$ is the set of homogeneous degree $k$ polynomials in $S=ℤ\left[{x}_{1},...,{x}_{n}\right]$ and ${\left({S}^{W}\right)}^{k}$ is the set of homogeneous degree $k$ polynomials in ${S}^{W}=ℤ\left[{e}_{1},...,{e}_{n}\right]=ℤ{\left[{x}_{1},...,{x}_{n}\right]}^{{S}_{n}}$ the Poincaré series of $S$ and ${S}^{W}$ are $1 (1-t)n = ∑ k≥0 dim(Sk)tk and ∏i=1n ( 1 1-ti ) = ∑ k≥0 dim((SW)k)tk.$ Then the Poincaré series of $S/I$ is $∏i=1n 1-ti 1-t = [n]! = 1⋅(1+t)⋯(1+t+⋯+tn-1).$ There are $n\left(n-1\right)\cdots 2\cdot 1=n!$ monomials in (???) and thus the monomials in (*) form a basis of $S$ as an ${S}^{W}$ module. The relations (???) provide a way to expand any polynomial in terms of this basis (with coefficients in ${S}^{W}$). $\square$

## The groups ${G}_{r,p,n}$

Let $r$ and $n$ be positive integers. The group ${G}_{r,1,n}$ is the group of $n×n$ matrices with

1. exactly one non zero entry in each row and each column,
2. the nonzero entries are ${r}^{\mathrm{th}}$ roots of 1.
Let $p$ be a positive integer (not necessarily prime) such that $p$ divides $r.$ The group ${G}_{r,p,n}$ is defined by the exact sequence ${1} → Gr,p,n → Gr,1,n →φ ℤ/pℤ → {1}, where φ(g) = ∏ gij≠0 gij p$ is the ${p}^{\mathrm{th}}$ power of the product of the nonzero entries of $g,$ and $ℤ/pℤ$ is identified with the group of ${p}^{\mathrm{th}}$ roots of unity. Thus ${G}_{r,p,n}=\mathrm{ker}\phi$ is a normal subgroup of ${G}_{r,1,n}$ of index $p.$ Examples are
1. ${G}_{1,1,n}={S}_{n}=W{A}_{n-1}$ is the symmetric group (the Weyl group of type ${A}_{n-1}$),
2. ${G}_{2,1,n}={O}_{n}\left(ℤ\right)=W{B}_{n}$ is the hyperoctahedral group of orthogonal matrices with entries in $ℤ$ (the Weyl group of type ${B}_{n}$),
3. ${G}_{2,2,n}=W{D}_{n}$ is the group of signed permutations with an even number of negative signs (the Weyl group of type ${D}_{n}$),
4. ${G}_{r,1,1}=ℤ/rℤ$ is the cyclic group of order $r$ of ${r}^{\mathrm{th}}$ roots of unity, and
5. ${G}_{r,r,2}=W{I}_{2}\left(r\right)$ is the dihedral group of order $2r.$

Let $\xi ={e}^{\frac{2\pi i}{r}}$ be the primitive ${r}^{\mathrm{th}}$ root of unity and let $𝔬=ℤ\left[\xi \right].$ If ${x}_{1},...,{x}_{n}$ is a basis of ${𝔬}^{n}$ then the natural action of ${G}_{r,p,n}$ extends uniquely to an action of ${G}_{r,p,n}$ on the polynomial ring $𝔬\left[{x}_{1},...,{x}_{n}\right]$ by ring automorphisms. The invariant ring is

Let

1. $𝔬{\left[{x}_{1},...,{x}_{n}\right]}^{{G}_{r,p,n}}=𝔬\left[{f}_{1},...,{f}_{n}\right].$
2. $𝔬\left[{x}_{1},...,{x}_{n}\right]$ is a free $𝔬{\left[{x}_{1},...,{x}_{n}\right]}^{{G}_{r,p,n}}-module$ with basis

 Proof. To show: ${f}_{1},...,{f}_{n}$ generate $𝔬{\left[{X}_{n}\right]}^{W}$ and they are algebraically independent. $\square$

Each element $w\in {G}_{r,1,n}$ can be written uniquely in the form $w= t1γ1⋯tnγnσ, where ti = diag(1,...,1,ξ,1,...,1), σ∈Sn, 0≤γi≤r-1,$ so that ${t}_{i}$ is the diagonal matrix with 1s on the diagonal except for $\xi$ in the ${i}^{\mathrm{th}}$ diagonal entry. The element and thus For each $w\in {G}_{r,p,n}$ define a monomial

The polynomial ring $𝔬\left[{x}_{1},...,{x}_{n}\right]$ is a free $𝔬{\left[{x}_{1},...,{x}_{n}\right]}^{{G}_{r,p,n}}-module$ with basis

 Proof. $\square$

## Notes and References

Taken from lecture notes on symmetric functions by Arun Ram.

References?