QSym over Sym has a stable basis Aaron Lauve and Sarah Mason 3 - - PowerPoint PPT Presentation

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis

QSym over Sym has a stable basis

Aaron Lauve and Sarah Mason 3 August 2010

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis

Symn ֒ → Q[x] free module coinvariant space Q[x]/Symn of dimension n!

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis

Symn ֒ → Q[x] free module coinvariant space Q[x]/Symn of dimension n! Symn ֒ → QSymn ֒ → Q[x]

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Symmetric polynomials in n variables f (x1, x2, . . . , xn) ∈ Symn ⇐ ⇒ πf = f ∀π ∈ Sn Example (Sym3) x1x2 + x1x3 + x2x3 x2

1x2 + x2 1x3 + x2 2x3

x2

1x2 + x1x2 2

x2

1x2 + x2 1x3 + x2 2x3 + x1x2 2 + x1x2 3 + x2x2 3

x3

1x2 + x2 3

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Partitions A partition of a positive integer n is a weakly decreasing sequence

• f positive integers which sum to n.

Example: 13 = 5 + 3 + 3 + 2 λ = (5, 3, 3, 2)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Semi-standard Young tableaux (SSYT) A semi-standard Young tableau is a filling of the entries in a partition diagram with positive integers so that the rows are weakly increasing from left to right and the columns are strictly increasing from top to bottom. The weight of a SSYT T, denoted xT is the product

i x# of i′s i

. Example: T = 1 2 2 4 7 3 4 7 4 6 9 7 8 xT = x1x2

2x3x3 4x6x3 7x8x9

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Schur function basis The Schur function sλ is the polynomial obtained by summing the weights of all SSYT of shape λ: sλ(x1, x2, . . . , xn) =

• T∈SSYT(λ)

xT s2,1(x1, x2, x3) = 1 1 2 1 1 3 1 2 2 1 2 3 1 3 2 1 3 3 2 2 3 2 3 3 x2

1x2 + x2 1x3 + x1x2 2 +

2x1x2x3 + x1x2

3 + x2 2x3 + x2x2 3

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Quasiymmetric polynomials in n variables f (x1, x2, . . . , xn) ∈ QSymn ⇐ ⇒ (∀i1 < i2 < . . . < ik) coeff of xa1

1 xa2 2 . . . xak k = coeff of xa1 i1 xa2 i2 · · · xak ik

Example (QSym3) x1x2 + x1x3 + x2x3 x2

1x2 + x2 1x3 + x2 2x3

x2

1x2 + x1x2 2

x2

1x2 + x2 1x3 + x2 2x3 + x1x2 2 + x1x2 3 + x2x2 3

x3

1x2 + x2 3

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Compositions A composition of a positive integer n is a sequence of positive integers which sum to n. Example: 14 = 2 + 4 + 1 + 4 + 3 α = (2, 4, 1, 4, 3)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Composition tableau (CT) A filling, F, of the cells of a composition such that:

1 The leftmost column entries strictly increase top to bottom. 2 The row entries weakly decrease from L to R. 3 The entries satisfy the triple rule: for i < j,

(F(j, k) = 0 and F(j, k) ≥ F(i, k)) = ⇒ F(j, k) > F(i, k − 1). F= 1 1 3 2 2 2 4 6 6 6 1 7 7 4 α = (2, 4, 1, 4, 3) xF = x3

1x3 2x3x2 4x3 6x2 7

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Quasisymmetric Schur function basis QSα =

• T∈CT(α)

xT, where CT(α) is the set of all composition tableau of shape α.

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

QS3,1,2(x1, x2, x3, x4) = 1 1 1 2 3 3 1 1 1 2 4 3 1 1 1 2 4 4 1 1 1 3 4 4 x3

1x2x2 3

+ x3

1x2x3x4

+ x3

1x2x2 4

+ x3

1x3x2 4

2 1 1 3 4 4 2 2 1 3 4 4 2 2 2 3 4 4 +x2

1x2x3x2 4

+ x1x2

2x3x2 4

+ x3

2x3x2 4

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

The quasisymmetric Schur functions: form a basis for all quasisymmetric functions refine the Schur functions in a natural way (sλ =

• ˜

α=λ

QSα) decompose into positive sums of Gessel’s fundamental quasisymemtric functions enjoy many nice combinatorial properties, including a Littlewood-Richardson style multiplication rule

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

QSα × sλ =

• β

QSβ A few definitions: contre-lattice A word is contre-lattice if the number of i′s in any prefix of the resulting word is greater than or equal to the number of (i − 1)’s in the prefix for all i. Example: 3 2 3 1 2 3 3 1 2, 3 2 1 2 1

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

β/α (def by example) β = (3, 1, 4, 2) α = (3, 2) β/α =                    , ,                   

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

reading word The column reading word (wcol) of a skew diagram is the word

• btained by reading the entries in the available cells down columns,

working from right to left. Littlewood-Richardson composition tableau (LRCT) A Littlewood-Richardson composition tableau (LRCT) is a composition tableau filling of a skew composition diagram with positive integers so that the column reading word is contre-lattice.

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Littlewood-Richardson composition tableaux (LRCT) 7 7 2 6 6 6 3 3 1 4 4 4 10 10 10 1 9 1 8 8 8 8 2 7 7 2 6 6 6 3 3 3 LRCT of shape LRCT of shape, (3,4,2,3)/(2,3,3) (4, 2, 5, 3, 6)/(3, 1, 4, 2, 3) wcol = 3213 wcol = 3231321

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Theorem (Haglund, Luoto, M, van Willigenburg 08) QSα · sλ =

• β

cβ

α,λQSβ,

where cβ

α,λ is the number of (LRCT) of shape in β/α with content

λ∗. Example (3 variables): QS1,2 · s1,1 = QS2,3 + QS1,1,3 + QS1,3,1 (2, 1) 5 1 4 4 2 6 1 4 4 2 5 4 4 2 1

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Symn ֒ → Q[x1, . . . , xn] = Q[x] A classical result The following are equivalent:

1 Symn is a polynomial ring, generated by the elementary

symmetric polynomials En = {e1(x), . . . , en(x)};

2 the ring Q[x] is a free Symn-module; 3 the coinvariant space Q[x]Sn = Q[x]/(En) has dimension n!

and is isomorphic to the regular representation of Sn.

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Symn ֒ → QSymn ֒ → Q[x] Bergeron-Reutenauer conjectures The ring QSymn is a free module over Symn; the dimension of the coinvariant space QSymn/(En) is n!; the set of pure and inverting compositions form the indexing set for a basis for QSymn/(En).

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Symn ֒ → QSymn ֒ → Q[x] Bergeron-Reutenauer conjectures The ring QSymn is a free module over Symn; Proved by Garsia-Wallach (2003) the dimension of the coinvariant space QSymn/(En) is n!; the set of pure and inverting compositions form the indexing set for a basis for QSymn/(En).

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Symn ֒ → QSymn ֒ → Q[x] Bergeron-Reutenauer conjectures The ring QSymn is a free module over Symn; Proved by Garsia-Wallach (2003) the dimension of the coinvariant space QSymn/(En) is n!; Proved by Garsia-Wallach (2003) the set of pure and inverting compositions form the indexing set for a basis for QSymn/(En).

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Symmetric polynomials Quasisymmetric polynomials The Bergeron-Reutenauer conjectures

Symn ֒ → QSymn ֒ → Q[x] Bergeron-Reutenauer conjectures The ring QSymn is a free module over Symn; Proved by Garsia-Wallach (2003) the dimension of the coinvariant space QSymn/(En) is n!; Proved by Garsia-Wallach (2003) the set of pure and inverting compositions form the indexing set for a basis for QSymn/(En). Lauve-M (2009)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Destandardization of permutations The destandardization d(σ) of a permutation σ is the lexicographically least word w whose standardization is σ. Example: If σ = 15243 then d(σ) = 13121 D(n) and Bn Let D(n) denote the set of all compositions {d(σ) : σ ∈ Sn}. Then Bergeron and Reutenauer define the sets Bn recursively so that B0 := {0}, 1n+Bn−1 ⊆ D(n) and D(n) is disjoint from Bn−1, and Bn := Bn−1 ∪ D(n) \

• 1n+Bn−1
• .

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

D(1) = { 1 } B0 = {0} D(2) = { 11 , 21} B1 = {0} D(3) = { 111 , 211, 121, 221, 212, 321 } B2 = {0, 21} D(4) = { 1111 , 2111, 1211, 1121, 2211, B3 = {0, 21, 211, 2121, 1221, 2112, 1212, 2221, 121, 221, 212} 2212, 2122, 3211 , 3121, 1321, 3221 , 2321 , 3212, 2312, 2132, 3321 , 3231 , 3213, 4321}

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Inverting compositions A composition γ is said to be inverting ⇐ ⇒ ∀i > 1 ∃ s < t such that γs = i and γt = i − 1. Example 4 1 3 2 2 1 3 4 1 2 2 1 3

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Pure compositions Factor γ so that γ = wkik . . . 2i21i1 where k is maximal such that w contains only entries greater than k. Then γ is said to be pure ⇐ ⇒ k is even. Example 4 3 4 2 2 1 1 w = 4 3 4, k = 2 (pure) 4 2 3 2 1 1 w = 4 2 3 2, k = 1 (impure) 4 1 3 2 2 1 3 w = 4 1 3 2 2 1 3, k = 0 (pure)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Bn = {pure and inverting compositions of length ≤ n } Examples B0 = {0} B1 = {0} B2 = {0, 21} B3 = {0, 21, 211, 121, 221, 212} B4 = {0, 21, 211, 121, 221, 212, 2111, 1211, 1121, 2211, 2121, 1221, 2112, 1212, 2221, 2212, 2122, 3121, 1321, 3212, 2312, 2132, 3213, 4321}

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Theorem (Lauve-M 2009) If Bn = {pure and inverting compositions of length ≤ n } then Bn = {QSβ|β ∈ Bn} is a basis for QSymn/(En).

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Break down by degree

Let QSymn,d be the collection of all quasisymmetric polynomials in n variables homogeneous of degree d. We prove that the collection An,d := {sλ · QSβ | β ∈ Bn, l(λ), l(β) ≤ n, |λ| + |β| = d} is a basis for QSymn,d. ⋆ Uses the combintorial formula (Haglund-Luoto-M-van Willigenburg) for the product of a quasisymmetric Schur function and a Schur function.

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The revlex order () on compositions

Let Cn,d be the set of compositions of d into at most n parts. For α ∈ Cn,d, we say α β iff either: λ(α) λ(β)

• r

λ(α) = λ(β) and α is lexicographically larger than β when read right to left. Example: d = 4, n = 3 4 13 31 22 112 121 211

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ =

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 5

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 4 5

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 4 5 4

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 4 4 5 4

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 4 2 4 5 4

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 4 2 4 5 2 4

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 1 4 2 4 5 2 4

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 1 4 2 1 4 5 2 4

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 1 4 2 1 1 4 5 2 4

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 1 4 2 1 1 4 5 2 4 1 1

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 1 4 2 1 1 4 5 2 4 1 1 φ(λ, β) : 3 8 5 2 2 7 9 4 7 1 1

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The leading term of sλ · QSβ

To obtain the index for the leading polynomial (in revlex order), simply add λi to the ith longest part of β. Call the resulting composition φ(λ, β). Example: s54442211111 · QS243113423 β = 2 4 3 1 1 3 4 2 3 λ = 1 4 2 1 1 4 5 2 4 1 1 φ(λ, β) : 3 8 5 2 2 7 9 4 7 1 1 s54442211111 · QS243113423 = QS38522794711+ smaller terms

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: The inverse of the map φ

Let α be an arbitrary composition of d into at most n parts and set (λ, β) := (∅, α).

1 If β is pure and inverting, then φ−1(α) := (λ, β) 2 If β is impure and inverting, then set

φ−1(α) := (λ + (1n), β − (1n)).

3

If β is not inverting, then let j be the smallest non-inverting part of β. Replace β by the composition obtained by subtracting 1 from each of the m parts greater than or equal to j and replace λ by the partition obtained by adding 1 to each of the first m parts.

4 Repeat until Step (1) or (2) above is followed.

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1 (3): 3 7 4 2 2 6 8 3 6 1 1 , λ = (111111)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1 (3): 3 7 4 2 2 6 8 3 6 1 1 , λ = (111111)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1 (3): 3 7 4 2 2 6 8 3 6 1 1 , λ = (111111) (3): 3 6 4 2 2 5 7 3 5 1 1 , λ = (222211)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1 (3): 3 7 4 2 2 6 8 3 6 1 1 , λ = (111111) (3): 3 6 4 2 2 5 7 3 5 1 1 , λ = (222211)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1 (3): 3 7 4 2 2 6 8 3 6 1 1 , λ = (111111) (3): 3 6 4 2 2 5 7 3 5 1 1 , λ = (222211) (3): 3 5 4 2 2 4 6 3 4 1 1 , λ = (333311)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1 (3): 3 7 4 2 2 6 8 3 6 1 1 , λ = (111111) (3): 3 6 4 2 2 5 7 3 5 1 1 , λ = (222211) (3): 3 5 4 2 2 4 6 3 4 1 1 , λ = (333311)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1 (3): 3 7 4 2 2 6 8 3 6 1 1 , λ = (111111) (3): 3 6 4 2 2 5 7 3 5 1 1 , λ = (222211) (3): 3 5 4 2 2 4 6 3 4 1 1 , λ = (333311) (3): 3 5 4 2 2 4 5 3 4 1 1 , λ = (433311)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1 (3): 3 7 4 2 2 6 8 3 6 1 1 , λ = (111111) (3): 3 6 4 2 2 5 7 3 5 1 1 , λ = (222211) (3): 3 5 4 2 2 4 6 3 4 1 1 , λ = (333311) (3): 3 5 4 2 2 4 5 3 4 1 1 , λ = (433311) (2): 2 4 3 1 1 3 4 2 3 , λ = (54442211111)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Example of φ−1 map

β = 3 8 5 2 2 7 9 4 7 1 1 (3): 3 7 4 2 2 6 8 3 6 1 1 , λ = (111111) (3): 3 6 4 2 2 5 7 3 5 1 1 , λ = (222211) (3): 3 5 4 2 2 4 6 3 4 1 1 , λ = (333311) (3): 3 5 4 2 2 4 5 3 4 1 1 , λ = (433311) (2): 2 4 3 1 1 3 4 2 3 , λ = (54442211111) φ−1(38522794711) = (54442211111 , 243113423)

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Proof: Upper triangularity of transition matrix

Since the leading terms of the polynomials in An,d are in bijective correspondence with composition of d into at most n parts, the transition matrix between the collection An,d and the quasisymmetric Schur function basis for QSymn,d is upper

• unitriangular. Therefore An,d is a basis for QSymn,d.

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Example (n = 3, d = 4)

QS4 QS13 QS31 QS22 QS112 QS121 QS211 s4 1 · · · · · · s31 · 1 1 · · · · s1 · QS21 · · 1 1 · · 1 s22 · · · 1 · · · s211 · · · · 1 · · QS121 · · · · · 1 · QS211 · · · · · · 1

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Corollary (Original Bergeron-Reutenauer basis) The set {Mβ|β ∈ Bn} is a basis for the Symn-module QSymn/(En). Proof. Expand the product sλMβ in the quasisymmetric Schur function

• basis. The leading term in this expansion is φ(λ, β).

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

The Hilbert series of a graded ring A is given by Hq(A) =

• k≥0

dkqk, where dk records the dimension of the kth graded piece of A. The Hilbert series for QSymn and Symn Hq(QSymn) = 1 + q 1 − q + · · · + qn (1 − q)n Hq(Symn) =

n

• i=1

1 1 − qi

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

Theorem (Bergeron-Reutenauer)

• c∈Bn

q|c| = Pn(q), where Pn(q) := Hq(QSymn) Hq(Symn) =

n−1

• i=1

(1+ q + q2 + . . . + qi)

n

• i=0

qi(1− q)n−i Therefore Pn(q) is the Hilbert series of the coinvariant space for quasisymmetric polynomials and Pn(1) = n! implies that the dimension of this space is n!.

Aaron Lauve and Sarah Mason QSym/Sym

Overview Symmetric and quasisymmetric polynomials The Bergeron-Reutenauer basis Garsia’s descent monomials Pure and inverting compositions A simple bijection Applications and corollaries

r-Qsym Is there a basis for r-Qsym which behaves like the quasisymmetric Schur polynomials? If so, what does multiplication by a Schur function do to these basis elements? Can we apply the same techniques to find combinatorial proofs of Garsia-Wallach results for r-Qsym/Sym?

Aaron Lauve and Sarah Mason QSym/Sym