6.3 Schur polynomials
6.3.1 Alternants
We work over a fixed positive integer \(N\) and the polynomial ring \(\mathcal{P} = \mathbb {Z}[x_1, x_2, \ldots , x_N]\). An \(N\)-partition is a weakly decreasing \(N\)-tuple \(\lambda = (\lambda _1, \lambda _2, \ldots , \lambda _N) \in \mathbb {N}^N\) with \(\lambda _1 \geq \lambda _2 \geq \cdots \geq \lambda _N\).
(a) We let \(\rho \) be the \(N\)-tuple \(\left( N-1,N-2,\ldots ,N-N\right) \in \mathbb {N}^{N}\).
(b) For any \(N\)-tuple \(\alpha =\left( \alpha _{1},\alpha _{2},\ldots ,\alpha _{N}\right) \in \mathbb {N}^{N}\), we define
This is called the \(\alpha \)-alternant (of \(x_{1},x_{2},\ldots ,x_{N}\)).
The \(\rho \) vector is strictly decreasing: if \(i {\lt} j\) (as indices in \(\{ 0, 1, \ldots , N{-}1\} \)), then \(\rho _i {\gt} \rho _j\).
Immediate from \(\rho _i = N - 1 - i\).
The sum of the \(\rho \) vector equals the triangular number:
Follows from \(\sum _{i=0}^{N-1}(N-1-i) = \sum _{k=0}^{N-1} k = N(N-1)/2\).
The \(\rho \) vector \((N{-}1, N{-}2, \ldots , 0)\) is itself an \(N\)-partition (since it is strictly decreasing, hence weakly decreasing).
- rhoPartition N = { parts := rhoVector N, antitone := ⋯ }
We have \(a_\rho = \prod _{1 \le i {\lt} j \le N}(x_i - x_j)\).
This is the classical Vandermonde determinant identity. The matrix for \(a_\rho \) has entry \((i,j)\) equal to \(x_i^{N-j}\), and its determinant equals \(\prod _{i{\lt}j}(x_i - x_j)\) by the Vandermonde formula.
Let \(\alpha \in \mathbb {N}^{N}\).
(a) If the \(N\)-tuple \(\alpha \) has two equal entries, then \(a_{\alpha }=0\).
(b) Let \(\beta \in \mathbb {N}^{N}\) be an \(N\)-tuple obtained from \(\alpha \) by swapping two entries. Then, \(a_{\beta }=-a_{\alpha }\).
(a) If \(\alpha _i = \alpha _j\), then two columns of the matrix \((x_k^{\alpha _\ell })\) are equal, so its determinant is \(0\).
(b) Swapping entries \(\alpha _i\) and \(\alpha _j\) amounts to permuting two columns of the matrix by a transposition, which multiplies the determinant by \(-1\).
6.3.2 Young diagrams and Schur polynomials
Let \(\lambda \) be an \(N\)-partition.
The Young diagram of \(\lambda \) is defined as the set
We visually represent each element \(\left( i,j\right) \) of this Young diagram as a box in row \(i\) and column \(j\).
We denote the Young diagram of \(\lambda \) by \(Y\left( \lambda \right) \).
- μ.youngDiagram = Finset.univ.biUnion fun (i : Fin N) => Finset.map { toFun := fun (j : ℕ) => (i, j), inj' := ⋯ } (Finset.range (μ.parts i))
A cell \((i,j)\) belongs to \(Y(\lambda )\) if and only if \(j {\lt} \lambda _i\).
Immediate from the definition.
Let \(\lambda \) be an \(N\)-partition.
A Young tableau of shape \(\lambda \) means a way of filling the boxes of \(Y\left( \lambda \right) \) with elements of \(\left[ N\right] \) (one element per box). Formally speaking, it is defined as a map \(T:Y\left( \lambda \right) \rightarrow \left[ N\right] \).
Let \(\lambda \) be an \(N\)-partition.
A Young tableau \(T\) of shape \(\lambda \) is said to be semistandard if its entries
increase weakly along each row (from left to right);
increase strictly down each column (from top to bottom).
Formally speaking, this means that a Young tableau \(T:Y\left( \lambda \right) \rightarrow \left[ N\right] \) is semistandard if and only if
we have \(T\left( i,j\right) \leq T\left( i,j+1\right) \) for any \(\left( i,j\right) \in Y\left( \lambda \right) \) satisfying \(\left( i,j+1\right) \in Y\left( \lambda \right) \);
we have \(T\left( i,j\right) {\lt}T\left( i+1,j\right) \) for any \(\left( i,j\right) \in Y\left( \lambda \right) \) satisfying \(\left( i+1,j\right) \in Y\left( \lambda \right) \).
We let \(\operatorname *{SSYT}\left( \lambda \right) \) denote the set of all semistandard Young tableaux of shape \(\lambda \).
- row_weak (i : Fin N) (j₁ j₂ : ℕ) : (i, j₂) ∈ lam.youngDiagram → j₁ < j₂ → self.entry (i, j₁) ≤ self.entry (i, j₂)
Entries increase weakly along rows
- col_strict (i₁ i₂ : Fin N) (j : ℕ) : (i₂, j) ∈ lam.youngDiagram → i₁ < i₂ → self.entry (i₁, j) < self.entry (i₂, j)
Entries increase strictly down columns
Let \(T\) be a semistandard Young tableau of shape \(\lambda \). If \((i, j_1)\) and \((i, j_2)\) are cells of \(Y(\lambda )\) with \(j_1 \le j_2\), then \(T(i,j_1) \le T(i,j_2)\).
If \(j_1 = j_2\) the claim is trivial; otherwise \(j_1 {\lt} j_2\) and we apply the row-weak condition.
Let \(T\) be a semistandard Young tableau of shape \(\lambda \). If \((i_1, j)\) and \((i_2, j)\) are cells of \(Y(\lambda )\) with \(i_1 \le i_2\), then \(T(i_1,j) \le T(i_2,j)\).
If \(i_1 = i_2\) the claim is trivial; otherwise \(i_1 {\lt} i_2\) and \(T(i_1,j) {\lt} T(i_2,j)\) by column-strictness.
In a semistandard tableau \(T\) of shape \(\lambda \), we have \(T(i,j) \ge i\) for every cell \((i,j) \in Y(\lambda )\).
By induction on \(i\). For \(i = 0\), this holds since entries are nonneg. For the inductive step, column-strictness gives \(T(i, j) {\gt} T(i{-}1, j) \ge i{-}1\), so \(T(i,j) \ge i\).
For any \(N\)-partition \(\lambda \), the tableau that fills every cell \((i,j)\) with the value \(i\) is semistandard. This is called the highest weight tableau.
Row-weakness holds because each row is constant. Column-strictness holds because entry values equal row indices, which are strictly increasing down columns.
The highest weight tableau achieves the minimum possible entry at each cell: for any semistandard tableau \(T\) of shape \(\lambda \) and any cell \(c \in Y(\lambda )\), the entry of the highest weight tableau at \(c\) is \(\le T(c)\).
The highest weight entry at \((i,j)\) is \(i\), and by Lemma 6.97, \(T(i,j) \ge i\).
The size of a Young tableau (the number of cells in \(Y(\lambda )\)) equals \(\sum _{i=1}^{N} \lambda _i\).
The Young diagram is the disjoint union of rows, and row \(i\) has \(\lambda _i\) cells.
Let \(\lambda \) be an \(N\)-partition. If \(T\) is any Young tableau of shape \(\lambda \), then we define the corresponding monomial
- T.monomial = ∏ c ∈ lam.youngDiagram, MvPolynomial.X (T.entry c)
For any Young tableau \(T\) of shape \(\lambda \),
Reindex the product over cells by partitioning according to the entry value.
Let \(\lambda \) be an \(N\)-partition. We define the Schur polynomial \(s_{\lambda }\in \mathcal{P}\) by
- schurPoly lam = ∑ f ∈ ssytFillingsYoung lam, fillingMonomialYoung f
6.3.3 Examples of Schur polynomials
The Schur polynomial \(s_{(n,0,\ldots ,0)}\) equals the complete homogeneous symmetric polynomial \(h_n\):
For a single-row partition \((n,0,\ldots ,0)\), the Young diagram has \(n\) cells in row 0. An SSYT filling is a weakly increasing sequence of \(n\) elements from \([N]\), which corresponds bijectively to a multiset of size \(n\). The monomials correspond under this bijection, so the sums are equal.
The Schur polynomial \(s_{(1^n, 0^{N-n})}\) (with \(n\) ones followed by zeros) equals the elementary symmetric polynomial \(e_n\):
For a single-column partition \((1^n)\), the Young diagram has \(n\) cells in column 0. An SSYT filling assigns strictly increasing values (by column-strictness), which corresponds bijectively to an \(n\)-element subset of \([N]\). The monomials match under this bijection.
For \(N \ge 2\), the Schur polynomial \(s_{(2,1,0,\ldots ,0)}\) equals \(e_2 \cdot e_1 - e_3\).
The Young diagram of \((2,1)\) has three cells: \((0,0)\), \((0,1)\), \((1,0)\). An SSYT filling assigns values \((i, j, k)\) with \(i \le j\) and \(i {\lt} k\). Summing \(x_i x_j x_k\) over all such triples and comparing with the expansion \(e_2 \cdot e_1 = 3 e_3 + \sum _{a{\lt}b} x_a^2 x_b + \sum _{a{\lt}b} x_a x_b^2\) yields the result.
6.3.4 Symmetry of Schur polynomials
Let \(\lambda \) be an \(N\)-partition. Then:
(a) The polynomial \(s_{\lambda }\) is symmetric.
(b) We have
Here, the addition on \(\mathbb {N}^{N}\) is defined entrywise.
(a) This follows from Theorem 6.127 applied to \(\mu = \mathbf{0}\), since \(s_{\lambda /\mathbf{0}} = s_\lambda \). The proof of Theorem 6.127 uses Bender–Knuth involutions.
(b) The proof uses Stembridge’s Lemma applied to \(\mu = 0\) and \(\nu = 0\). The only \(0\)-Yamanouchi semistandard tableau of shape \(\lambda \) is the highest weight tableau \(T_0\) (where each row \(i\) contains only \(i\)’s). The content of \(T_0\) equals \(\lambda \), so the result reduces to \(a_\rho \cdot s_\lambda = a_{\lambda + \rho }\).
6.3.5 Skew Young diagrams and skew Schur polynomials
Let \(\lambda \) and \(\mu \) be two \(N\)-partitions.
We say that \(\mu \subseteq \lambda \) if and only if \(Y\left( \mu \right) \subseteq Y\left( \lambda \right) \). Equivalently, \(\mu \subseteq \lambda \) if and only if
Thus we have defined a partial order \(\subseteq \) on the set of all \(N\)-partitions.
- NPartition.instLE = { le := fun (μ ν : NPartition N) => ∀ (i : Fin N), μ.parts i ≤ ν.parts i }
We have \(\mu \le \lambda \) if and only if \(Y(\mu ) \subseteq Y(\lambda )\).
\((\Rightarrow )\): If \(\mu _i \le \lambda _i\) for all \(i\) and \((i,j) \in Y(\mu )\) (i.e. \(j {\lt} \mu _i\)), then \(j {\lt} \lambda _i\), so \((i,j) \in Y(\lambda )\). \((\Leftarrow )\): If \(\mu _i {\gt} \lambda _i\) for some \(i\), then \((i, \lambda _i) \in Y(\mu ) \setminus Y(\lambda )\).
Let \(\lambda \) and \(\mu \) be two \(N\)-partitions such that \(\mu \subseteq \lambda \). Then, we define the skew Young diagram \(Y\left( \lambda /\mu \right) \) to be the set difference
- skewYoungDiagram lam mu = lam.youngDiagram \ mu.youngDiagram
A cell \((i,j)\) belongs to \(Y(\lambda /\mu )\) if and only if \(\mu _i \le j {\lt} \lambda _i\) (in 0-indexed coordinates).
Follows from the definition \(Y(\lambda /\mu ) = Y(\lambda ) \setminus Y(\mu )\) and the membership characterization of Young diagrams.
The skew Young diagram \(Y(\lambda /\mathbf{0})\) equals the full Young diagram \(Y(\lambda )\).
Since \(\mathbf{0}_i = 0\) for all \(i\), the condition \(\mu _i \le j\) is automatic, and the skew diagram reduces to the full diagram.
Let \(\lambda \) and \(\mu \) be two \(N\)-partitions such that \(\mu \subseteq \lambda \). Let \(\left( a,b\right) \) and \(\left( e,f\right) \) be two elements of \(Y\left( \lambda /\mu \right) \). Let \(\left( c,d\right) \in \mathbb {Z}^{2}\) satisfy \(a\leq c\leq e\) and \(b\leq d\leq f\). Then, \(\left( c,d\right) \in Y\left( \lambda /\mu \right) \).
We need \(\mu _c \le d\) and \(d {\lt} \lambda _c\). Since \(\mu \) is weakly decreasing and \(a \le c\), we have \(\mu _c \le \mu _a \le b \le d\). Since \(\lambda \) is weakly decreasing and \(c \le e\), we have \(d \le f {\lt} \lambda _e \le \lambda _c\).
Let \(\lambda \) and \(\mu \) be two \(N\)-partitions such that \(\mu \subseteq \lambda \). A Young tableau of shape \(\lambda /\mu \) means a way of filling the boxes of \(Y\left( \lambda /\mu \right) \) with elements of \(\left[ N\right] \) (one element per box). Formally speaking, it is defined as a map \(T:Y\left( \lambda /\mu \right) \rightarrow \left[ N\right] \).
Young tableaux of shape \(\lambda /\mu \) are often called skew Young tableaux.
If we don’t have \(\mu \subseteq \lambda \), then we agree that there are no Young tableaux of shape \(\lambda /\mu \).
Let \(\lambda \) and \(\mu \) be two \(N\)-partitions.
A Young tableau \(T\) of shape \(\lambda /\mu \) is said to be semistandard if its entries
increase weakly along each row (from left to right);
increase strictly down each column (from top to bottom).
Formally speaking, this means that a Young tableau \(T:Y\left( \lambda /\mu \right) \rightarrow \left[ N\right] \) is semistandard if and only if
we have \(T\left( i,j\right) \leq T\left( i,j+1\right) \) for any \(\left( i,j\right) \in Y\left( \lambda /\mu \right) \) satisfying \(\left( i,j+1\right) \in Y\left( \lambda /\mu \right) \);
we have \(T\left( i,j\right) {\lt}T\left( i+1,j\right) \) for any \(\left( i,j\right) \in Y\left( \lambda /\mu \right) \) satisfying \(\left( i+1,j\right) \in Y\left( \lambda /\mu \right) \).
We let \(\operatorname *{SSYT}\left( \lambda /\mu \right) \) denote the set of all semistandard Young tableaux of shape \(\lambda /\mu \).
- row_weak (i : Fin N) (j₁ j₂ : ℕ) : (i, j₁) ∈ skewYoungDiagram lam mu → (i, j₂) ∈ skewYoungDiagram lam mu → j₁ < j₂ → self.entry (i, j₁) ≤ self.entry (i, j₂)
Entries increase weakly along rows
- col_strict (i₁ i₂ : Fin N) (j : ℕ) : (i₁, j) ∈ skewYoungDiagram lam mu → (i₂, j) ∈ skewYoungDiagram lam mu → i₁ < i₂ → self.entry (i₁, j) < self.entry (i₂, j)
Entries increase strictly down columns
Let \(\lambda \) and \(\mu \) be two \(N\)-partitions. Let \(T\) be a semistandard Young tableau of shape \(\lambda /\mu \). Then:
(a) If \(\left( i,j_{1}\right) \) and \(\left( i,j_{2}\right) \) are two elements of \(Y\left( \lambda /\mu \right) \) satisfying \(j_{1}\leq j_{2}\), then \(T\left( i,j_{1}\right) \leq T\left( i,j_{2}\right) \).
(b) If \(\left( i_{1},j\right) \) and \(\left( i_{2},j\right) \) are two elements of \(Y\left( \lambda /\mu \right) \) satisfying \(i_{1}\leq i_{2}\), then \(T\left( i_{1},j\right) \leq T\left( i_{2},j\right) \).
(c) If \(\left( i_{1},j\right) \) and \(\left( i_{2},j\right) \) are two elements of \(Y\left( \lambda /\mu \right) \) satisfying \(i_{1}{\lt}i_{2}\), then \(T\left( i_{1},j\right) {\lt}T\left( i_{2},j\right) \).
(d) If \(\left( i_{1},j_{1}\right) \) and \(\left( i_{2},j_{2}\right) \) are two elements of \(Y\left( \lambda /\mu \right) \) satisfying \(i_{1}\leq i_{2}\) and \(j_{1}\leq j_{2}\), then \(T\left( i_{1},j_{1}\right) \leq T\left( i_{2},j_{2}\right) \).
(e) If \(\left( i_{1},j_{1}\right) \) and \(\left( i_{2},j_{2}\right) \) are two elements of \(Y\left( \lambda /\mu \right) \) satisfying \(i_{1}{\lt}i_{2}\) and \(j_{1}\leq j_{2}\), then \(T\left( i_{1},j_{1}\right) {\lt}T\left( i_{2},j_{2}\right) \).
(a) If \(j_1 = j_2\) the claim is trivial; otherwise apply row-weakness.
(b) If \(i_1 = i_2\) the claim is trivial; otherwise \(T(i_1,j) {\lt} T(i_2,j)\) by column-strictness.
(c) This is just the column-strict condition.
(d) By convexity (Lemma 6.113), \((i_2, j_1) \in Y(\lambda /\mu )\). Then \(T(i_1, j_1) \le T(i_2, j_1) \le T(i_2, j_2)\) using parts (b) and (a).
(e) Similarly, \((i_2, j_1) \in Y(\lambda /\mu )\) by convexity. Then \(T(i_1, j_1) {\lt} T(i_2, j_1) \le T(i_2, j_2)\) using parts (c) and (a).
Let \(\lambda \) and \(\mu \) be two \(N\)-partitions. If \(T\) is any Young tableau of shape \(\lambda /\mu \), then we define the corresponding monomial
- T.monomial = ∏ c ∈ skewYoungDiagram lam mu, MvPolynomial.X (T.entry c)
For any skew Young tableau \(T\) of shape \(\lambda /\mu \),
Reindex the product over cells by partitioning according to the entry value.
The skew Young diagram \(Y(\lambda /\lambda )\) is empty for any \(N\)-partition \(\lambda \).
By the membership characterization, \((i,j) \in Y(\lambda /\lambda )\) iff \(\lambda _i \le j {\lt} \lambda _i\), which is impossible.
For a skew Young tableau \(T\) of shape \(\lambda /\mu \), if \(j {\lt} \mu _i\) then \(T(i,j) = 0\). That is, entries to the left of the inner partition are zero.
If \(j {\lt} \mu _i\), then \((i,j) \notin Y(\lambda /\mu )\), so the support condition gives \(T(i,j) = 0\).
For a skew Young tableau \(T\) of shape \(\lambda /\mu \), if \(j \ge \lambda _i\) then \(T(i,j) = 0\). That is, entries to the right of the outer partition are zero.
If \(j \ge \lambda _i\), then \((i,j) \notin Y(\lambda /\mu )\), so the support condition gives \(T(i,j) = 0\).
6.3.6 Filling infrastructure for Schur polynomials
The formal definition of Schur polynomials represents SSYT as functions from the diagram to \([N]\) (“fillings”), filters for those satisfying the semistandard conditions, and sums the associated monomials.
A filling of a skew Young diagram \(Y(\lambda /\mu )\) is a function \(f : Y(\lambda /\mu ) \to [N]\). A filling is semistandard if it satisfies the row-weak and column-strict conditions. The filling monomial is \(x_f = \prod _{c \in Y(\lambda /\mu )} x_{f(c)}\).
- One or more equations did not get rendered due to their size.
- ssytFillings lam mu = Finset.filter (isSSYTFilling lam mu) Finset.univ
- fillingMonomial f = ∏ c : { c : Fin N × ℕ // c ∈ skewYoungDiagram lam mu }, MvPolynomial.X (f c)
The content of a filling \(f\) of \(Y(\lambda /\mu )\) is the function \(\mathrm{cont}(f) : [N] \to \mathbb {N}\) defined by
For any filling \(f\) of \(Y(\lambda /\mu )\),
Reindex the product over cells by partitioning according to the entry value (fiberwise product decomposition).
Let \(\lambda \) and \(\mu \) be two \(N\)-partitions. We define the skew Schur polynomial \(s_{\lambda /\mu }\in \mathcal{P}\) by
- skewSchurPoly lam mu = ∑ f ∈ ssytFillings lam mu, fillingMonomial f
For any \(N\)-partition \(\lambda \), we have \(s_{\lambda /\mathbf{0}} = s_\lambda \).
The semistandard tableaux of shape \(\lambda /\mathbf{0}\) are precisely the semistandard tableaux of shape \(\lambda \), since \(Y(\lambda /\mathbf{0}) = Y(\lambda )\) by Lemma 6.112.
Let \(\lambda \) and \(\mu \) be any two \(N\)-partitions. Then, the polynomial \(s_{\lambda /\mu }\) is symmetric.
The proof uses the Bender–Knuth involutions. For each \(k \in \{ 0, 1, \ldots , N{-}2\} \), the \(k\)-th Bender–Knuth involution \(\mathrm{BK}_k\) is a bijection on \(\operatorname {SSYT}(\lambda /\mu )\) satisfying:
\(\mathrm{BK}_k\) is an involution: \(\mathrm{BK}_k \circ \mathrm{BK}_k = \mathrm{id}\);
\(\mathrm{BK}_k\) preserves the shape of the tableau;
\(x_{\mathrm{BK}_k(T)} = s_k \cdot x_T\) (where \(s_k\) swaps \(x_k\) and \(x_{k+1}\)).
Since the simple transpositions \(s_0, s_1, \ldots , s_{N-2}\) generate \(S_N\), and each \(\mathrm{BK}_k\) establishes that \(s_k \cdot s_{\lambda /\mu } = s_{\lambda /\mu }\) (Lemma 6.137), we conclude that \(s_{\lambda /\mu }\) is symmetric.
The reduction from arbitrary permutations to adjacent transpositions uses Lemma 6.138 (strong induction on the distance \(|j - i|\)).
6.3.7 Bender–Knuth involutions (helper lemmas)
The following helper lemmas formalize the key properties of the Bender–Knuth involution, used in the proof of Theorem 6.127.
The cell bijection between two representations of skew Young diagrams. The first uses 0-indexed columns (\((i,j) \in Y(\lambda /\mu )\) iff \(\mu _i \le j {\lt} \lambda _i\)), and the second uses 1-indexed columns (\((i,j) \in Y(\lambda /\mu )\) iff \(\mu _i {\lt} j \le \lambda _i\)). The bijection is \((i, j) \mapsto (i, j+1)\).
The filling bijection converts fillings between the two skew diagram representations by composing with the cell bijection.
- skewFillingEquiv lam mu = (skewCellEquiv lam mu).arrowCongr (Equiv.refl (Fin N))
The filling bijection preserves the semistandard property: a filling \(f\) is semistandard if and only if its image under the filling bijection is semistandard in the 1-indexed representation.
Both row-weak and column-strict conditions depend only on the relative ordering of indices, which is preserved by the column shift \(j \mapsto j+1\).
The filling bijection preserves content: for any filling \(f\) and index \(i\), the content of \(f\) at \(i\) equals the content of the corresponding filling in the 1-indexed representation at \(i\).
The cell bijection induces a bijection between \(\{ c : f(c) = i\} \) and the corresponding set in the 1-indexed representation, since the entry values are unchanged.
The Bender–Knuth involution \(\mathrm{BK}_k\) on fillings of \(Y(\lambda /\mu )\). For a semistandard filling \(f\), \(\mathrm{BK}_k(f)\) swaps certain entries \(k\) and \(k{+}1\) while preserving semistandardness. For non-semistandard fillings, it returns the input unchanged.
- One or more equations did not get rendered due to their size.
The Bender–Knuth involution \(\mathrm{BK}_k\) preserves SSYT membership: if \(f\) is a semistandard filling, then so is \(\mathrm{BK}_k(f)\).
Transfers through the filling bijection: the image is semistandard by the Bender–Knuth semistandard preservation theorem, and the bijection preserves semistandardness.
The Bender–Knuth map is an involution: applying it twice returns the original filling.
Transfers through the filling bijection using the Bender–Knuth involutivity theorem.
Let \(f\) be a semistandard filling and \(f' = \mathrm{BK}_k(f)\). Then the content of \(f'\) at index \(k\) equals the content of \(f\) at \(k{+}1\), the content of \(f'\) at \(k{+}1\) equals the content of \(f\) at \(k\), and the content at all other indices is unchanged.
This is the key property of the Bender–Knuth involution. Proved via the Bender–Knuth content swap theorem, transferred through the filling bijection using Lemma 6.131.
The monomial effect of the Bender–Knuth involution: \(x_{\mathrm{BK}_k(f)} = (\text{swap } x_k\, x_{k+1}) \cdot x_f\).
This says that \(\mathrm{BK}_k\) effectively swaps the roles of variables \(x_k\) and \(x_{k+1}\) in the monomial.
By Lemma 6.135, the content of \(\mathrm{BK}_k(f)\) is obtained from that of \(f\) by swapping entries \(k\) and \(k{+}1\). Since the monomial is \(\prod _i x_i^{\mathrm{cont}(f)(i)}\), this transposition of the content corresponds to renaming \(x_k \leftrightarrow x_{k+1}\).
The skew Schur polynomial is invariant under simple transpositions:
for each \(k \in \{ 0, 1, \ldots , N{-}2\} \).
Apply \(s_k\) to the sum \(s_{\lambda /\mu } = \sum _f x_f\). By Lemma 6.136, \(s_k \cdot x_f = x_{\mathrm{BK}_k(f)}\). Since \(\mathrm{BK}_k\) is an involution on the SSYT fillings (Lemmas 6.133 and 6.134), reindexing the sum by \(\mathrm{BK}_k\) shows \(\sum _f x_{\mathrm{BK}_k(f)} = \sum _f x_f\).
If a polynomial \(P\) is invariant under every simple transposition \(s_k\) (swapping \(x_k\) and \(x_{k+1}\)), then \(P\) is invariant under every transposition (swapping any two variables \(x_i\) and \(x_j\)).
By strong induction on the distance \(|j - i|\). If \(|j - i| = 1\), the swap is already a simple transposition. Otherwise, decompose \(\mathrm{swap}(i, j) = \mathrm{swap}(k, j) \cdot \mathrm{swap}(i, k) \cdot \mathrm{swap}(k, j)\) where \(k = j - 1\), and apply the inductive hypothesis to the smaller swap.
Renaming variables in a filling monomial equals applying the permutation to each entry:
Follows from \(\mathrm{rename}(\sigma )(X_i) = X_{\sigma (i)}\) applied to each factor of the product.
6.3.8 Equivalences between Schur polynomial definitions
The project maintains multiple representations of SSYT and Schur polynomials for different proof contexts. This subsection records the key equivalence results.
The two definitions of Schur polynomials in this project (one using \(\mathbb {Z}\) coefficients and \(N\)-partitions, the other using generic coefficients and raw \(N\)-tuples) are equal.
Both definitions sum over semistandard Young tableaux of shape \(\lambda \). The cell bijection \((i,j) \mapsto (i,j+1)\) between the two indexing conventions preserves the SSYT conditions and the monomials.
An equivalence between the two SSYT representations in this project:
The first uses a total function \(T : [N] \times \mathbb {N} \to [N]\) with a support condition (entries outside the diagram are \(0\)).
The second uses a dependent function where the column index is bounded by the partition part: \(T(i, j)\) for \(j {\lt} \lambda _i\).
The equivalence converts between the total function and the bounded representations.
- SSYTEquiv.ssytEquiv lam = { toFun := SSYTEquiv.toSchurBasicsSSYT, invFun := SSYTEquiv.toSFSSYT, left_inv := ⋯, right_inv := ⋯ }
The two Schur polynomial definitions agree over \(\mathbb {Z}\). More precisely, the Schur polynomial defined via fillings of the Young diagram equals the one defined via the bounded representation.
Both are sums over SSYT with the same monomials. The equivalence on tableaux preserves the monomial at each tableau, so the sums are equal.
Corollary of Theorem 6.142: the filling-based Schur polynomial equals the bounded-representation Schur polynomial (with appropriate conversion of the partition).
Direct application of Theorem 6.142 with the partition conversion being an involution.