2.1 Gröbner theory

Let ${\mathbf {x}} = (x_1, \dots , x_N)$ be a finite list of variables, and let ${\mathbb {F}}[{\mathbf {x}}_N]$ be the polynomial ring in these variables over a field ${\mathbb {F}}$ . A total order $<$ on the monomials in ${\mathbb {F}}[{\mathbf {x}}_N]$ is a term order if

• • we have $1 \leq m$ for all monomials m, and

• • if $m_1, m_2, m_3$ are monomials with $m_1 \leq m_2$ , then $m_1 m_3 \leq m_2 m_3$ .

If $f \in {\mathbb {F}}[{\mathbf {x}}_N]$ is a nonzero polynomial and $<$ is a term order, write ${\mathrm {in}}_< (f)$ for the largest monomial with respect to $<$ which appears with nonzero coefficient in f.

Let $I \subseteq {\mathbb {F}}[{\mathbf {x}}_N]$ be an ideal, and let $<$ be a term order. The initial ideal ${\mathrm {in}}_< (I) \subseteq {\mathbb {F}}[{\mathbf {x}}_N]$ associated to I is given by

(2.1) $$ \begin{align} {\mathrm {in}}_< (I) := \langle {\mathrm {in}}_< (f) \,:\, f \in I, \, \, f \neq 0 \rangle \subseteq {\mathbb {F}}[{\mathbf {x}}_N]. \end{align} $$

In other words, the ideal ${\mathrm {in}}_< (I)$ is generated by the $<$ -leading monomials of all nonzero polynomials in I. A subset $G = \{ g_1, \dots , g_r \} \subseteq I$ is a Gröbner basis of I if

(2.2) $$ \begin{align} {\mathrm {in}}_< (I) = \langle {\mathrm {in}}_< (g_1), \dots, {\mathrm {in}}_< (g_r) \rangle. \end{align} $$

If $G = \{ g_1, \dots , g_r \}$ is a Gröbner basis of I, it follows that $I = \langle g_1, \dots , g_r \rangle $ .

Given an ideal $I \subseteq {\mathbb {F}}[{\mathbf {x}}_N]$ and a term order $<$ , a monomial m in the variables ${\mathbf {x}}_N$ is a standard monomial if $m \neq {\mathrm {in}}_< (f)$ for any nonzero $f \in I$ . It is known that the family of cosets

(2.3) $$ \begin{align} \{ m + I \,:\, m\ \text{a standard monomial} \} \end{align} $$

descends to a vector space basis of ${\mathbb {F}}[{\mathbf {x}}_N]/I$ . This is referred to as the standard monomial basis.

2.2 Orbit harmonics

Let $Z \subseteq {\mathbb {F}}^N$ be a finite locus of points, and consider the ideal

(2.4) $$ \begin{align} {\mathbf {I}}(Z) := \{ f \in {\mathbb {F}}[{\mathbf {x}}_N] \,:\, f({\mathbf {z}}) = 0 \text{ for all } {\mathbf {z}} \in Z \} \end{align} $$

of polynomials in ${\mathbb {F}}[{\mathbf {x}}_N]$ which vanish on Z. The ideal ${\mathbf {I}}(Z)$ is usually not homogeneous. Since Z is finite, we have an identification

(2.5) $$ \begin{align} {\mathbb {F}}[Z] \cong {\mathbb {F}}[{\mathbf {x}}_N]/{\mathbf {I}}(Z) \end{align} $$

of the vector space ${\mathbb {F}}[Z]$ of functions $Z \rightarrow {\mathbb {F}}$ and the typically ungraded quotient space ${\mathbb {F}}[{\mathbf {x}}_N]/{\mathbf {I}}(Z)$ .

Given a nonzero polynomial $f \in {\mathbb {F}}[{\mathbf {x}}_N]$ , let $\tau (f)$ be the highest degree homogeneous component of f. That is, if $f = f_d + \cdots + f_1 + f_0$ where $f_i$ is homogeneous of degree i and $f_d \neq 0$ , we have $\tau (f) = f_d$ . If $I \subseteq {\mathbb {F}}[{\mathbf {x}}_N]$ is an ideal, the associated graded ideal is

(2.6) $$ \begin{align} {\mathrm {gr}} \, I := \langle \tau(f) \,:\, f \in I, \, \, f \neq 0 \rangle. \end{align} $$

In other words, the ideal ${\mathrm {gr}} \, I$ is generated by the top homogeneous components of all nonzero polynomials in I. The associated graded ideal ${\mathrm {gr}} \, I \subseteq {\mathbb {F}}[{\mathbf {x}}_N]$ is homogeneous by construction.

Returning to the setting of our finite locus $Z \subseteq {\mathbb {F}}^N$ , we may extend the chain (2.5) of ungraded ${\mathbb {F}}$ -vector space isomorphisms

(2.7) $$ \begin{align} {\mathbb {F}}[Z] \cong {\mathbb {F}}[{\mathbf {x}}_N] / {\mathbf {I}}(Z) \cong {\mathbb {F}}[{\mathbf {x}}_N] / {\mathrm {gr}} \, {\mathbf {I}}(Z), \end{align} $$

where the last quotient ${\mathbb {F}}[{\mathbf {x}}_N] / {\mathrm {gr}} \, {\mathbf {I}}(Z)$ has the additional structure of a graded ${\mathbb {F}}$ -vector space.

When the locus Z possesses symmetry, more can be said. Let $G \subseteq GL_N({\mathbb {F}})$ be a finite matrix group, and assume that the group algebra ${\mathbb {F}}[G]$ is semisimple. Equivalently, this means that $|G| \neq 0$ in ${\mathbb {F}}$ . The natural action of G on ${\mathbb {F}}^N$ induces an action on ${\mathbb {F}}[{\mathbf {x}}_N]$ by linear substitutions. If Z is stable under the action of G, the isomorphisms (2.7) hold in the category of ${\mathbb {F}}[G]$ -modules, and the last quotient ${\mathbb {F}}[{\mathbf {x}}_N] / {\mathrm {gr}} \, {\mathbf {I}}(Z)$ has the additional structure of a graded ${\mathbb {F}}[G]$ -module.

2.3 The Schensted correspondence

Given $n \geq 0$ , a partition of n is a weakly decreasing sequence $\lambda = (\lambda _1 \geq \cdots \geq \lambda _k)$ of positive integers which satisfy $\lambda _1 + \cdots + \lambda _k = n$ . We write $\lambda \vdash n$ to indicate that $\lambda $ is a partition of n. We identify a partition $\lambda = (\lambda _1, \dots , \lambda _k)$ with its (English) Young diagram consisting of $\lambda _i$ left-justified boxes in row i.

Let $\lambda \vdash n$ be a partition. A tableau of shape $\lambda $ is an assignment $T: \lambda \rightarrow \{1,2,\dots \}$ of positive integers to the boxes of $\lambda $ . A standard tableau of shape $\lambda $ is a bijective filling $T: \lambda \rightarrow [n]$ of the boxes of $\lambda $ with $1, 2, \dots , n$ which is increasing across rows and down columns. We display, from left to right, the Young diagram of $\lambda = (4,2,2) \vdash 8$ , a standard tableau of shape $\lambda $ and two tableaux of shape $\lambda $ which are not standard.

Although the above tableau $T: \lambda \rightarrow \{1,2, \dots \}$ on the far right is not standard, it is an injective filling which is (strictly) increasing across rows and down columns. We call a tableau satisfying these conditions a partial standard tableau.

The famous Schensted correspondence [Reference Schensted18] is a bijection

(2.8) $$ \begin{align} {\mathfrak {S}}_n \xrightarrow{ \quad \sim \quad} \bigsqcup_{\lambda \vdash n} \{ (P, Q) \,:\, P, Q \in {\mathrm {SYT}}(\lambda) \} \end{align} $$

which sends a permutation $w \in {\mathfrak {S}}_n$ to a pair $(P(w), Q(w))$ of standard tableaux with the same n-box shape. The Schensted correspondence is most commonly defined using an insertion algorithm (see, e.g., [Reference Sagan17] for details). We will not need the insertion formulation of the Schensted bijection, but an equivalent ‘geometric’ formulation due to Viennot [Reference Viennot, Foata and Notes20] recalled in the next section will be crucial in our work. Schensted proved that his bijection relates to increasing subsequences as follows.

2.4 ${\mathfrak {S}}_n$ -representation theory

Let ${\mathbb {F}}$ be a field in which $n \neq 0$ so that the group algebra ${\mathbb {F}}[{\mathfrak {S}}_n]$ is semisimple. There is a one-to-one correspondence between partitions of n and irreducible representations of ${\mathfrak {S}}_n$ over ${\mathbb {F}}$ . If $\lambda \vdash n$ is a partition, we write $V^\lambda $ for the corresponding irreducible module, $\chi ^\lambda : {\mathfrak {S}}_n \rightarrow {\mathbb {F}}$ for its character, and $f^\lambda := \dim V^\lambda $ for its dimension. The number $f^\lambda $ counts standard tableaux of shape $\lambda $ .

The vector space ${\mathrm {Class}}({\mathfrak {S}}_n,{\mathbb {F}})$ of ${\mathbb {F}}$ -valued class functions on ${\mathfrak {S}}_n$ has basis $\{ \chi ^\lambda \,:\, \lambda \vdash n \}$ given by irreducible characters. The Kronecker product $*$ on ${\mathrm {Class}}({\mathfrak {S}}_n,{\mathbb {F}})$ is defined by

(2.9) $$ \begin{align} (\varphi * \psi)(w) := \varphi(w) \cdot \psi(w) \end{align} $$

for any $\varphi , \psi \in {\mathrm {Class}}({\mathfrak {S}}_n,{\mathbb {F}})$ and $w \in {\mathfrak {S}}_n$ . If $V_1$ and $V_2$ are ${\mathfrak {S}}_n$ -modules, their vector space tensor product $V_1 \otimes V_2$ carries a diagonal action of ${\mathfrak {S}}_n$ by the rule $w \cdot (v_1 \otimes v_2) := (w \cdot v_1) \otimes (w \cdot v_2)$ . The characters $\chi _{V_1}, \chi _{V_2}, \chi _{V_1 \otimes V_2}: {\mathfrak {S}}_n \rightarrow {\mathbb {F}}$ of these modules are related by $\chi _{V_1 \otimes V_2} = \chi _{V_1} * \chi _{V_2}$ .

3.1 The injection relations

In order to analyze the quotients ${\mathbb {F}}[{\mathbf {x}}_{n \times n}] / I_n$ , we start by exhibiting strategic elements of the ideal $I_n$ . Given two subsets $S, T \subseteq [n]$ , define elements $a_{S,T}, b_{S,T} \in {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ by

(3.1) $$ \begin{align} a_{S,T} := \sum_{f: S \hookrightarrow T} \left( \prod_{ i \in S } x_{i,f(i)} \right) \quad \quad \text{and} \quad \quad b_{S,T} := \sum_{f: S \hookrightarrow T} \left( \prod_{ i \in S } x_{f(i),i} \right), \end{align} $$

where both sums are over injective functions $f: S \hookrightarrow T$ . For example, if $S = \{2,4\}$ and $T = \{1,3,4\}$ , we have

$$ \begin{align*} a_{S,T} & = x_{2,1} x_{4,3} + x_{2,1} x_{4,4} + x_{2,3} x_{4,1} + x_{2,3} x_{4,4} + x_{2,4} x_{4,1} + x_{2,4} x_{4,3}, \\b_{S,T} & = x_{1,2} x_{3,4} + x_{1,2} x_{4,4} + x_{3,2} x_{1,4} + x_{3,2} x_{4,4} + x_{4,2} x_{1,4} + x_{4,2} x_{3,4}. \end{align*} $$

In general, the polynomials $a_{S,T}$ and $b_{S,T}$ are obtained from one another by transposing the matrix ${\mathbf {x}}_{n \times n}$ of variables. We have $a_{S,T} = b_{S,T} = 0$ whenever $|S|> |T|$ .

Since the product of any two variables in the same row or column of ${\mathbf {x}}_{n \times n}$ is a generator of $I_n$ , we have the congruences

(3.2) $$ \begin{align} a_{S,T} \equiv \prod_{i \in S} \left( \sum_{j \in T} x_{i,j} \right) \quad\mod I_n \quad \quad \text{and} \quad \quad b_{S,T} \equiv \prod_{i \in S} \left( \sum_{j \in T} x_{j,i} \right) \quad\mod I_n \end{align} $$

modulo $I_n$ . In other words, as far as the quotient ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ is concerned, we could have defined $a_{S,T}$ and $b_{S,T}$ using all functions $S \rightarrow T$ , not just injections. Our first lemma states that $a_{S,T}$ and $b_{S,T}$ are members of $I_n$ provided that $|S| + |T|> n$ .

Proof. The polynomial $b_{S,T}$ is obtained from $a_{S,T}$ by transposing the matrix ${\mathbf {x}}_{n \times n}$ of variables, an operation under which $I_n$ is stable. As such, it suffices to prove the lemma for $a_{S,T}$ . Furthermore, by the congruences (3.2) it suffices to prove the lemma when $|S| + |T| = n+1$ . Finally, since $I_n$ is stable under the action of the product group ${\mathfrak {S}}_n \times {\mathfrak {S}}_n$ on the rows and columns of ${\mathbf {x}}_{n \times n}$ , it is enough to consider the case where $S = [s]$ and $T = [t]$ for $s + t = n+1$ .

We argue by increasing induction on s (and decreasing induction on t). If $s = 1$ , then $t = n$ and $a_{S,T} = x_{1,1} + x_{1,2} + \cdots + x_{1,n}$ is a generator of the ideal $I_n$ . If $s> 1$ , we have

(3.3) $$ \begin{align} a_{S,T} \equiv \prod_{i = 1}^s \left( \sum_{j = 1}^t x_{i,j} \right) = (x_{1,1} + x_{1,2} + \cdots + x_{1,t}) \times \left[ \prod_{i = 2}^s \left( \sum_{j = 1}^{t+1} x_{i,j} \right) \right] - {\mathbf {E}}, \end{align} $$

where the congruence modulo $I_n$ follows from Equation (3.2), the expression $[ \, \cdots ]$ in square brackets lies in $I_n$ by induction and the ‘error term’ ${\mathbf {E}}$ is given by

(3.4) $$ \begin{align} {\mathbf {E}} = (x_{1,1} + x_{1,2} + \cdots + x_{1,t}) \times \sum_{\varnothing \neq S' \subseteq \{2, \dots, s\}} \left( \prod_{i' \in S'} x_{i',t+1} \times \prod_{i \in \{2,\dots,s\} - S'} (x_{i,1} + \cdots + x_{i,t}) \right). \end{align} $$

It suffices to show that ${\mathbf {E}} \in I_n$ . When the $|S'|> 1$ and $i^{\prime }_1, i^{\prime }_2 \in S'$ are distinct, the corresponding summand in ${\mathbf {E}}$ contains the product $x_{i^{\prime }_1,t+1} \cdot x_{i^{\prime }_2,t+1}$ and so lies in $I_n$ . We conclude that

(3.5) $$ \begin{align} {\mathbf {E}} \equiv (x_{1,1} + x_{1,2} + \cdots + x_{1,t}) \times \sum_{i_0 = 2}^s \left( x_{i_0,t+1} \times \prod_{2 \leq i \leq s}^{i \neq i_0} (x_{i,1} + \cdots + x_{i,t}) \right) \end{align} $$

modulo $I_n$ . Applying the congruences (3.2) and the defining relations of $I_n$ , we arrive at

(3.6) $$ \begin{align} {\mathbf {E}} \equiv \pm (x_{1,t+1} + x_{1,t+2} + \cdots + x_{1,n}) \times \sum_{i_0 = 2}^s \left( x_{i_0,t+1} \times \prod_{2 \leq i \leq s}^{i \neq i_0} (x_{i,t+2} + x_{i,t+3} \cdots + x_{i,n}) \right). \end{align} $$

The sum $(x_{i,t+2} + x_{i,t+3} \cdots + x_{i,n}) $ contains $n-t-1 = n - (n+1-s) - 1 = s - 2$ terms. The pigeonhole principle implies that every term in the expansion of the right-hand side of the congruence (3.6) will contain a product of variables $x_{i,j} \cdot x_{i',j}$ for some $i \neq i'$ so that ${\mathbf {E}} \in I_n$ . We conclude that $a_{S,T} \in I_n$ , and the lemma is proven.

3.2 Shadow sets

We represent a permutation $w = [w(1), \dots , w(n)] \in {\mathfrak {S}}_n$ with its graph, that is, the collection of points $\{ (i,w(i)) \,:\, 1 \leq i \leq n \}$ on the grid $[n] \times [n]$ . For example, the permutation $w = [4,1,8,5,3,6,2,7] \in {\mathfrak {S}}_8$ is given below in bullets.

Viennot used [Reference Viennot, Foata and Notes20] the graph of a permutation w to obtain its image $(P(w),Q(w))$ under the Schensted correspondence as follows. Shine a flashlight northeast from the origin (0,0). Each bullet in the permutation casts a shadow to its northeast. The boundary of the shaded region is the first shadow line; in our example, it is as follows.

Removing the points on the first shadow line and repeating this procedure, we obtain the second shadow line. Iterating, we obtain the third shadow line, the fourth shadow line and so on. In our example, the shadow lines are shown below.

Let $w \in {\mathfrak {S}}_n$ and suppose that the shadow lines of w are given by $L_1, \dots , L_r$ from southwest to northeast. Viennot proved [Reference Viennot, Foata and Notes20] that if $w \mapsto (P(w), Q(w))$ under the Schensted correspondence, then the y-coordinates of the infinite horizontal rays in $L_1, \dots , L_r$ form the first row of $P(w)$ and the x-coordinates of the infinite vertical rays of $L_1, \dots , L_r$ form the first row of $Q(w)$ . In our example, the first row of $P(w)$ is while the first row of $Q(w)$ is . In particular, the common length of the first row of $P(w)$ and $Q(w)$ is the number of shadow lines. The northeast corners of the shadow lines played an important role in Viennot’s work and will for us as well.

In our example, the points in the shadow set ${\mathcal {S}}(w) = \{ (2,4), (4,8), (5,5), (7,3) \}$ are drawn in red. For any permutation $w \in {\mathfrak {S}}_n$ , the shadow set ${\mathcal {S}}(w)$ contains at most one point in any row or column. Such subsets of the square grid have a name.

Rook placements are also known as ‘partial permutations’. Importantly, the Viennot shadow line construction may be performed on an arbitrary rook placement, not just on the graph of a permutation.

Although every permutation shadow set is a rook placement, not every rook placement is the shadow set of a permutation. For example, shadow sets contain no points in row 1 or column 1. In Lemma 3.6 below, we give a combinatorial criterion for deciding whether a rook placement is a shadow set.

Returning to our permutation $w \in {\mathfrak {S}}_n$ , we may iterate the shadow line construction on the shadow set ${\mathcal {S}}(w)$ . In our $n = 8$ example, this yields the shadow lines.

Viennot proved that the horizontal and vertical rays of these ‘iterated’ shadow lines give the second rows of $P(w)$ and $Q(w)$ , respectively. In our example, the second row of $P(w)$ is and the second row of $Q(w)$ is . These iterated shadow lines produce an iterated shadow set ${\mathcal {S}}({\mathcal {S}}(w))$ whose points are drawn in blue. Repeating this procedure in our example yields the iterated shadow sets and shadow lines

and we conclude that the tableaux $P(w)$ and $Q(w)$ are given by

respectively.

For our purposes, we may take Theorem 3.4 as the definition of the Schensted correspondence. Combining Theorem 3.4 with Schensted’s Theorem 2.1 yields the following result immediately.

Lemma 3.5. Let $w \in {\mathfrak {S}}_n$ . The size $|{\mathcal {S}}(w)|$ of the shadow set of w is given by

(3.7) $$ \begin{align} |{\mathcal {S}}(w)| = n - {\mathrm {lis}}(w). \end{align} $$

We close this subsection with a combinatorial criterion for deciding when a rook placement ${\mathcal {R}}$ is the shadow set of some permutation $w \in {\mathfrak {S}}_n$ . We use the fact that the shadow line construction may be applied to ${\mathcal {R}}$ . This will yield a pair $(P,Q)$ of partial standard tableaux with the same shape such that the y-coordinates of ${\mathcal {R}}$ are the entries in P and the x-coordinates in ${\mathcal {R}}$ are the entries in Q.

Lemma 3.6. Let ${\mathcal {R}} \subseteq [n] \times [n]$ be a rook placement, and apply the shadow line construction to ${\mathcal {R}}$ . Let $L_1, \dots , L_r$ be the shadow lines so obtained. Define two length n sequences $x_1 x_2 \dots x_n$ and $y_1 y_2 \dots y_n$ over the alphabet $\{1,0,-1\}$ by

(3.8) $$ \begin{align} x_i = \begin{cases} 1 & \text{if one of the lines } L_1, \dots, L_r \text{ has a vertical ray at } x = i, \\ -1 & \text{ if the vertical line } x = i \text{ does not meet } {\mathcal {R}}, \\ 0 & \text{otherwise.} \end{cases} \end{align} $$

and

(3.9) $$ \begin{align} y_i = \begin{cases} 1 & \text{if one of the lines } L_1, \dots, L_r \text{ has a horizontal ray at } y = i, \\ -1 & \text{if the horizontal line } y = i \text{ does not meet } {\mathcal {R}}, \\ 0 & \text{otherwise.} \end{cases} \end{align} $$

Then ${\mathcal {R}} = {\mathcal {S}}(w)$ is the shadow set of some permutation $w \in {\mathfrak {S}}_n$ if and only if for all $1 \leq i \leq n$ we have $x_1 + x_2 + \cdots + x_i \leq 0$ and $y_1 + y_2 + \cdots + y_i \leq 0$ .

An example may help in understanding Lemma 3.6 and its proof. Let $n = 8$ , and let ${\mathcal {R}}$ be the rook placement

$$ \begin{align*} {\mathcal {S}} = \{ (2,8), (3,7), (5,3), (6,5), (7,6) \} \end{align*} $$

of size 5. Applying the Viennot shadow line construction to ${\mathcal {R}}$ yields

where the sequences $x_1 x_2 \dots x_8$ and $y_1 y_2 \cdots y_8$ in $\{1,0,-1\}$ are shown horizontally and vertically, respectively. A $+1$ in a given row (or column) corresponds to an infinite ray of a shadow line, a 0 corresponds to a shadow line segment which is not an infinite ray, and a $-1$ corresponds to that row (or column) not containing an element of ${\mathcal {R}}$ . We have $x_1 + x_2 + \cdots + x_7 = 1> 0$ , so by Lemma 3.6 the set ${\mathcal {R}}$ is not the shadow set of a permutation in ${\mathfrak {S}}_8$ . Indeed, applying Schensted insertion to the rook placement ${\mathcal {R}}$ yields the pair of tableaux $P'$ and $Q'$ given by

(respectively), and adding the row corresponding to the positions of the $-1$ ’s in the sequence $x_1 x_2 \dots x_8$ to the top row of $Q'$ would not yield a standard tableau.

3.3 Shadow monomials and spanning

Our next task is to convert the combinatorics of the previous subsection into a spanning set for the quotient ring ${\mathbb {F}}[{\mathbf {x}}_{n \times n}] / I_n$ . Given any set ${\mathcal {S}} \subseteq [n] \times [n]$ of grid points, let $m({\mathcal {S}}) = \prod _{(i,j) \in {\mathcal {S}}} x_{i,j}$ be the corresponding squarefree monomial in ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ .

The spanning set of Lemma 3.7 is far from a basis. In order to extract a basis from this spanning set, we introduce a strategic term order. Recall that the lexicographical order on monomials in an ordered set of variables $y_1> y_2 > \cdots > y_N$ is given by $y_1^{a_1} \cdots y_N^{a_N} < y_1^{b_1} \cdots y_N^{b_N}$ if there exists $1 \leq j \leq N$ with $a_i = b_i$ for $i < j$ and $a_j < b_j$ .

Definition 3.8. The Toeplitz term order $<_{\mathrm {Top}}$ on monomials in ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ is the lexicographical term order with respect to the order on variables given by

(3.10) $$ \begin{align} x_{1,1}> x_{2,1} > x_{1,2} > x_{3,1} > x_{2,2} > x_{1,3} > \cdots > x_{n,n-1} > x_{n-1,n} > x_{n,n}. \end{align} $$

Roughly speaking, the Toeplitz term order weights a variable $x_{a,b}$ heavier than $x_{c,d}$ whenever $a + b < c + d$ and then breaks ties lexicographically. In fact, this tie breaking process among variables $x_{i,j}$ with $i + j$ constant will be irrelevant for the arguments that follow; all that is important is the relative weight of the variables $x_{i,j}$ for which $i + j$ differs. The word ‘Toeplitz’ comes from Toeplitz matrices (which are constant along diagonals). Since all of the relations we apply will be homogeneous, we could have also defined $<_{\mathrm {Top}}$ by ordering by total degree first and then using the lexicographical order with respect to the indicated variable order to break ties.

For example, if $w = [4,1,8,5,3,6,2,7] \in {\mathfrak {S}}_8$ we have ${\mathcal {S}}(w) = \{ (2,4), (4,8), (5,5), (7,3) \}$ so that ${\mathfrak {s}}(w) = x_{2,4} \cdot x_{4,8} \cdot x_{5,5} \cdot x_{7,3}$ . Our next lemma shows that the shadow monomials of permutations span the quotient ring ${\mathbb {F}}[{\mathbf {x}}_{n \times n}] / I_n$ . The key tools in its proof are the relations in ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ coming from Lemma 3.1 and the characterization (Lemma 3.6) of when a rook placement monomial $m({\mathcal {R}})$ is the shadow monomial ${\mathfrak {s}}(w)$ of a permutation $w \in {\mathfrak {S}}_n$ . To begin, we record the $<_{\mathrm {Top}}$ -leading terms of the elements of $I_n$ appearing in Lemma 3.1.

Observation 3.10. Let $S = \{s_1 < \cdots < s_p \}$ and $T = \{t_1 < \cdots < t_q \}$ be subsets of $[n]$ with $p \leq q$ . Then

(3.11) $$ \begin{align} {\mathrm {in}}_{<_{\mathrm {Top}}}(a_{S,T}) = x_{s_1,t_1} x_{s_2,t_2} \cdots x_{s_p,t_p} \quad \quad \text{and} \quad \quad {\mathrm {in}}_{<_{\mathrm {Top}}}(b_{S,T}) = x_{t_1,s_1} x_{t_2,s_2} \cdots x_{t_p,s_p}. \end{align} $$

In other words, the leading monomials of $a_{S,T}$ and $b_{S,T}$ correspond to the injection $S \hookrightarrow T$ which assigns the elements of S to the smallest $|S|$ elements of T in an order-preserving fashion. If $S = \{2,4\}$ and $T = \{1,4,5\}$ , then $a_{S,T}$ given by

$$ \begin{align*} a_{S,T} = \underline{x_{2,1} x_{4,4}} + x_{2,4} x_{4,1} + x_{2,1} x_{4,5} + x_{2,5} x_{4,1} + x_{2,4} x_{4,5} + x_{2,5} x_{4,4} \end{align*} $$

with its $<_{\mathrm {Top}}$ -leading term underlined. We have all the pieces we need to prove our spanning result.

Proof. Let ${\mathcal {R}} \subseteq [n] \times [n]$ be a rook placement. By Lemma 3.7, it suffices to show that $m({\mathcal {R}})$ lies in the span of $\{ {\mathfrak {s}}(w) \,:\, w \in {\mathfrak {S}}_n \}$ modulo $I_n$ . If ${\mathcal {R}} = {\mathcal {S}}(w)$ for some permutation $w \in {\mathfrak {S}}_n$ , then $m({\mathcal {R}}) = {\mathfrak {s}}(w)$ and this is clear, so assume that ${\mathcal {R}} \neq {\mathcal {S}}(w)$ for all $w \in {\mathfrak {S}}_n$ .

Apply Viennot’s shadow line construction to the rook placement ${\mathcal {R}}$ . Let $L_1, \dots , L_r$ be the shadow lines so obtained, ordered from southwest to northeast, and let $x_1 x_2 \dots x_n$ and $y_1 y_2 \dots y_n$ be the sequences appearing in the statement of Lemma 3.6. Since ${\mathcal {R}}$ is not the shadow set of a permutation, Lemma 3.6 implies that at least one of the sequences $x_1 x_2 \dots x_n$ and $y_1 y_2 \dots y_n$ has a prefix with a strictly positive sum. We assume that $x_1 x_2 \dots x_n$ has a prefix with strictly positive sum; the case of $y_1 y_2 \dots y_n$ is similar.

Choose $1 \leq a \leq n$ minimal such that $x_1 + x_2 + \cdots + x_a> 0$ . By the minimality of a, we have $x_a = 1$ so that $x = a$ is the vertical ray of one of the shadow lines $L_p$ for some $1 \leq p \leq r$ . We define a size p subset $\{ (i_1, j_1), \dots , (i_p, j_p) \} \subseteq {\mathcal {R}}$ as follows. Starting at the vertical ray of $L_p$ , let $(i_p,j_p)$ be the first element of ${\mathcal {R}}$ encountered by marching south (in particular, we have $i_p = a$ ). Now, march west from $(i_p,j_p)$ until one encounters a vertical segment of the shadow line $L_{p-1}$ . March south along this segment until one reaches a point $(i_{p-1},j_{p-1}) \in {\mathcal {R}}$ . Now, march west from $(i_{p-1},j_{p-1})$ until one encounters a vertical segment of the shadow line $L_{p-2}$ . March south along this segment until one reaches a point $(i_{p-2}, j_{p-2}) \in {\mathcal {R}}$ . Continuing this process, we arrive at a subset $\{ (i_1, j_1), \dots , (i_p, j_p) \} \subseteq {\mathcal {R}}$ such that

• • the point $(i_q, j_q)$ lies on the shadow line $L_q$ for each $1 \leq q \leq p$ ,

• • we have $i_1 < \cdots < i_p$ , and

• • we have $j_1 < \cdots < j_p$ .

Let ${\mathcal {R}}' := {\mathcal {R}} - \{ (i_1, j_1), \dots , (i_p, j_p) \}$ be the complement of $\{ (i_1, j_1), \dots , (i_p, j_p) \}$ in ${\mathcal {R}}$ .

An example may help in understanding these constructions. Let $n = 11$ and consider the rook placement ${\mathcal {R}} \subseteq [11] \times [11]$ given by

$$ \begin{align*}{\mathcal {R}} = \{ (2,9), (3,8), (4,3), (6,2), (7,6), (8,7), (9,5), (11,11) \}. \end{align*} $$

The sequence $(x_1, x_2, \dots , x_{11})$ is given by

$$ \begin{align*}(x_1, x_2, \dots, x_{11}) = (-1,1,0,0,-1,0,1,1,0,-1,1);\end{align*} $$

the figure below shows the shadow lines of ${\mathcal {R}}$ . By Lemma 3.6, the rook placement ${\mathcal {R}}$ is not the shadow set of a permutation in ${\mathfrak {S}}_8$ because

$$ \begin{align*}x_1 + x_2 + \cdots + x_8 = 1> 0.\end{align*} $$

Furthermore, the prefix $x_1 x_2 \dots x_8$ is the shortest positive sum prefix of the word $x_1 x_2 \dots x_{11}$ . We conclude that $a = 8$ . Our marching procedure on the shadow line diagram of ${\mathcal {R}}$ is shown in dashed and blue as follows.

We conclude that $(i_1, j_1) = (4,3), (i_2, j_2) = (7,6)$ and $(i_3, j_3) = (8,7)$ . Furthermore, we have the set

$$ \begin{align*} {\mathcal {R}}' = {\mathcal {R}} - \{ (i_1,j_1), (i_2,j_2), (i_3,j_3) \} = \{ (2,9), (3,8), (6,2), (9,5), (11,11) \} \end{align*} $$

of rooks in ${\mathcal {R}}$ which are not visited by the dashed blue line.

Consider the squarefree monomial $m({\mathcal {R}}')$ corresponding to the rooks in ${\mathcal {R}}' \subseteq {\mathcal {R}}$ which are not reached by our marching procedure. The ideal $m({\mathcal {R}}') \cdot {\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ generated by $m({\mathcal {R}}')$ in the ring ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ admits a morphism from a smaller quotient of the same form. More precisely, let $\bar {n} := n - |{\mathcal {R}}'|$ , and let $\bar {{\mathbf {x}}}$ be the $\bar {n} \times \bar {n}$ matrix of variables

(3.12) $$ \begin{align} \bar{{\mathbf {x}}} = \{ x_{i,j} \,:\, \text{neither the vertical line}\ x = i\ \text{nor the horizontal line}\ y = j\ \text{meet the set}\ {\mathcal {R}}' \}. \end{align} $$

In our example above, the matrix $\bar {{\mathbf {x}}}$ consists of the variables $x_{i,j}$ indexed by $i \in \{1,4,5,7,8,10\}$ and $j \in \{1,3,4,6,7,10\}$ . Let ${\mathbb {F}}[\bar {{\mathbf {x}}}]$ be the polynomial ring over the variables in $\bar {{\mathbf {x}}}$ , and let $\bar {I} \subseteq {\mathbb {F}}[\bar {{\mathbf {x}}}]$ be the natural copy of the ideal $I_{\bar {n}}$ in the square variable matrix $\bar {{\mathbf {x}}}$ . The map

(3.13) $$ \begin{align} \varphi: {\mathbb {F}}[\bar{{\mathbf {x}}}] / \bar{I} \longrightarrow m({\mathcal {R}}') \cdot {\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n \end{align} $$

induced by $f \mapsto m({\mathcal {R}}') \cdot f$ is easily seen to be a (well-defined) homomorphism of ${\mathbb {F}}[\bar {{\mathbf {x}}}]$ -modules; one simply checks that for any generator $g \in {\mathbb {F}}[\bar {{\mathbf {x}}}]$ of $\bar {I}$ , we have $m({\mathcal {R}}') \cdot g \in I_n$ . We consider the sets

(3.14) $$ \begin{align} T := \{ i_1 < i_2 < \cdots < i_p < i_p + 1 < i_p + 2 < \cdots < n \} - \{ i \,:\, (i,j) \in {\mathcal {R}}' \text{ for some} j \} \end{align} $$

and

(3.15) $$ \begin{align} S := \{ j_1 < j_2 < \cdots < j_p \}. \end{align} $$

In our example, we have $T = \{4,7,8,10\}$ and $S = \{3,6,7\}$ .

By the definitions of S and T, the polynomial $b_{S,T} \in {\mathbb {F}}[\bar {{\mathbf {x}}}]$ does not involve any of the variables which share a row or column with a rook $(i,j) \in {\mathcal {R}}'$ which is not visited by our marching procedure. Since $i_p = a$ and we have the prefix inequality $x_1 + x_2 + \cdots + x_a> 0$ , we have $|S| + |T|> \bar {n}$ . Lemma 3.1 applies to give

(3.16) $$ \begin{align} b_{S,T} \in \bar{I}. \end{align} $$

Since the map $\varphi $ of Equation (3.13) is a homomorphism of ${\mathbb {F}}[\bar {{\mathbf {x}}}]$ -modules we obtain

(3.17) $$ \begin{align} \varphi(b_{S,T}) = m({\mathcal {R}}') \cdot b_{S,T} \in I_n. \end{align} $$

Observation 3.10 implies that the Toeplitz-leading term of $m({\mathcal {R}}') \cdot b_{S,T}$ is $m({\mathcal {R}})$ , so the membership (3.17) yields

(3.18) $$ \begin{align} m({\mathcal {R}}) \equiv \Sigma \quad\mod I_n, \end{align} $$

where $\Sigma $ is a ${\mathbb {F}}$ -linear combination of monomials which are $<_{\mathrm {Top}} m({\mathcal {R}})$ . By induction on the Toeplitz order, the lemma is proven.

Lemma 3.11 (and its proof) give a Gröbner basis for the ideal $I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ with respect to the Toeplitz order which consists of

• • any product of two variables in ${\mathbf {x}}_{n \times n}$ which lie in the same row or column, and

• • in the notation of the proof of Lemma 3.11 and polynomial of the form $m({\mathcal {R}}') \cdot b_{S,T}$ for a rook placement ${\mathcal {R}} \subseteq [n] \times [n]$ which is not the shadow set of a permutation $w \in {\mathfrak {S}}_n$ for which some prefix of the word $x_1 x_2 \dots x_n$ is positive, and the image of $m({\mathcal {R}}') \cdot b_{S,T}$ under the involution ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ which interchanges $x_{i,j}$ and $x_{j,i}$ .

This Gröbner basis is far from minimal. We leave the computation of a minimal (or reduced) Gröbner basis of $I_n$ as an open problem.

3.4 Standard monomial basis and Hilbert series

Lemma 3.11 bounds the quotient ring ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ from above by giving an ${\mathbb {F}}$ -linear spanning set. In this subsection, we use orbit harmonics to bound this quotient from below.

Let ${\mathbb {F}}^{n \times n}$ be the affine space of $n \times n$ matrices over ${\mathbb {F}}$ with coordinate ring ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ . Write $P_n \subseteq {\mathbb {F}}^{n \times n}$ for the locus of permutation matrices. That is, the set $P_n$ consists of 0,1-matrices with a unique 1 in each row and column. The vanishing ideal ${\mathbf {I}}(P_n) \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ of the permutation matrix locus is generated by

• • $x_{i,j}^2 - x_{i,j}$ for all $1 \leq i, j \leq n$ ,

• • $x_{i,j} \cdot x_{i',j}$ for all $1 \leq i < i' \leq n$ and $i \leq j \leq n$ ,

• • $x_{i,j} \cdot x_{i,j'}$ for all $1 \leq i \leq n$ and $1 \leq j < j' \leq n$ ,

• • $x_{i,1} + \cdots + x_{i,n} - 1$ for all $1 \leq i \leq n$ , and

• • $x_{1,j} + \cdots + x_{n,j} - 1$ for all $1 \leq j \leq n$ .

Indeed, the generators in the first bullet point come from the $(i,j)$ -entry of a permutation matrix being 0 or 1, the generators in the second and third bullet points come from products of distinct entries in a row or column of a permutation matrix vanishing, and the generators in the fourth and fifth bullet points come from the row and columns summing to 1. Comparing these generators with Definition 1.1, we get the containment

(3.19) $$ \begin{align} I_n \subseteq {\mathrm {gr}} \, {\mathbf {I}}(P_n). \end{align} $$

Although the highest degree components $\tau (g_1), \dots , \tau (g_r)$ of a generating set $\{ g_1, \dots , g_r \}$ of an ideal I are in general insufficient to generate ${\mathrm {gr}} \, I$ , in our case the containment (3.19) is an equality.

Standard monomial bases of quotient rings ${\mathbb {F}}[{\mathbf {x}}]/I$ can be unpredictable, even for nicely presented ideals I. However, Theorem 3.12 informally suggests that the Toeplitz term order $<_{\mathrm {Top}}$ and the homogeneous ideal $I_n$ ‘know’ the Viennot shadow line incarnation of the Schensted correspondence $w \mapsto (P(w), Q(w))$ .

Proof. The chain (2.7) of ${\mathbb {F}}$ -vector space isomorphisms coming from orbit harmonics reads

(3.20) $$ \begin{align} {\mathbb {F}}[P_n] \cong {\mathbb {F}}[{\mathbf {x}}_{n \times n}] / {\mathbf {I}}(P_n) \cong {\mathbb {F}}[ {\mathbf {x}}_{n \times n} ] / {\mathrm {gr}} \, {\mathbf {I}}(P_n). \end{align} $$

Lemma 3.11 and the containment (3.19) of ideals yield the chain of (in)equalities

(3.21) $$ \begin{align} n! = | P_n | = \dim {\mathbb {F}}[ {\mathbf {x}}_{n \times n} ] / {\mathrm {gr}} \, {\mathbf {I}}(P_n) \leq \dim {\mathbb {F}}[{\mathbf {x}}_{n \times n}] / I_n \leq n! \end{align} $$

which forces $I_n = {\mathrm {gr}} \, {\mathbf {I}}(P_n)$ and $\dim {\mathbb {F}}[{\mathbf {x}}_{n \times n}] / I_n = n!$ . Another application of Lemma 3.11 shows that the spanning set $\{ {\mathfrak {s}}(w) \,:\, w \in {\mathfrak {S}}_n \}$ of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ is in fact a basis. The proof of Lemma 3.11 shows that $\{ {\mathfrak {s}}(w) \,:\, w \in {\mathfrak {S}}_n \}$ is the standard monomial basis of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ with respect to $<_{\mathrm {Top}}$ .

As a corollary, we get our promised relationship between the Hilbert series of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ and longest increasing subsequences in permutations.

Corollary 3.13. Let $a_{n,k}$ be the number of permutations in ${\mathfrak {S}}_n$ whose longest increasing sequence has length k. The quotient ring ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ has Hilbert series

(3.22) $$ \begin{align} {\mathrm {Hilb}}( {\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n; q) = a_{n,n} + a_{n,n-1} \cdot q + \cdots + a_{n,1} \cdot q^{n-1}. \end{align} $$

3.5 Local permutation statistics

Corollary 3.13 gives the structure of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ as a graded vector space. Our next goal is the structure of this quotient as a graded ${\mathfrak {S}}_n \times {\mathfrak {S}}_n$ module (at least when $n! \neq 0$ in ${\mathbb {F}}$ ). Our calculation of the module structure of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ will make crucial use of a notion of complexity on permutation statistics due to Hamaker and the author [Reference Hamaker and Rhoades10] called ‘locality’.

A permutation statistic (with values in the field ${\mathbb {F}}$ ) is a function $f: {\mathfrak {S}}_n \rightarrow {\mathbb {F}}$ . The study of permutation statistics is an important subfield of combinatorics. Examples include the exceedance, inversion, and peak numbers given by

(3.23) $$ \begin{align} {\mathrm {exc}}(w) &:= | \{ 1 \leq i \leq n \,:\, w(i)> i \} | \end{align} $$

(3.24) $$ \begin{align} {\mathrm {inv}}(w) &:= | \{ 1 \leq i < j \leq n \,:\, w(i)> w(j) \} | \end{align} $$

(3.25) $$ \begin{align} {\mathrm {peak}}(w) &:= | \{ 1 < i < n \,:\, w(i-1) < w(i)> w(i+1) \} |. \end{align} $$

Following [Reference Hamaker and Rhoades10], we define a notion of locality for permutation statistics as follows. If ${\mathcal {R}} \subseteq [n] \times [n]$ is a rook placement and $w \in {\mathfrak {S}}_n$ is a permutation, we say that w extends ${\mathcal {R}}$ if we have the containment of sets ${\mathcal {R}} \subseteq \{ (i,w(i)) \,:\, 1 \leq i \leq n \}$ . Given a rook placement ${\mathcal {R}} \subseteq [n] \times [n]$ , let ${\mathbf {1}}_{{\mathcal {R}}}: {\mathfrak {S}}_n \rightarrow {\mathbb {F}}$ be the indicator permutation statistic

(3.26) $$ \begin{align} {\mathbf {1}}_{{\mathcal {R}}}(w) = \begin{cases} 1 & \text{if } w \text{ extends } {\mathcal {R}}, \\ 0 & \text{otherwise,} \end{cases} \end{align} $$

which detects whether w extends ${\mathcal {R}}$ . A permutation statistic $f: {\mathfrak {S}}_n \rightarrow {\mathbb {F}}$ is k-local if there exist field elements $c_{{\mathcal {R}}} \in {\mathbb {F}}$ such that

(3.27) $$ \begin{align} f = \sum_{|{\mathcal {R}}| \, = \, k} c_{{\mathcal {R}}} \cdot {\mathbf {1}}_{{\mathcal {R}}} \end{align} $$

as functions ${\mathfrak {S}}_n \rightarrow {\mathbb {F}}$ where the sum is over all rook placements ${\mathcal {R}} \subseteq [n] \times [n]$ with k rooks.

Roughly speaking, the locality of a permutation statistic bounds its complexity. The only 0-local statistics are constant functions ${\mathfrak {S}}_n \rightarrow {\mathbb {F}}$ . The statistic ${\mathrm {exc}}$ is 1-local, the statistic ${\mathrm {inv}}$ is 2-local, and the statistic ${\mathrm {peak}}$ is 3-local. Following Hamaker and the author [Reference Hamaker and Rhoades10], we consider the ${\mathbb {F}}$ -vector space

(3.28) $$ \begin{align} {\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}}) := \{ f: {\mathfrak {S}}_n \rightarrow {\mathbb {F}} \,:\, f \ \text{is}\ k\text{-local} \} \end{align} $$

of k-local statistics on ${\mathfrak {S}}_n$ . It is not hard to see that any k-local statistic is also $(k+1)$ -local so that ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}}) \subseteq {\mathrm {Loc}}_{k+1}({\mathfrak {S}}_n,{\mathbb {F}})$ . Furthermore, any permutation statistic ${\mathfrak {S}}_n \rightarrow {\mathbb {F}}$ is $(n-1)$ -local. The vector spaces ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}})$ will play an important role in the module structure of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ (Theorem 4.2); for now we use shadow monomials to solve an open problem from [Reference Hamaker and Rhoades10] about the spaces ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}})$ themselves.

By definition, the set $\{ {\mathbf {1}}_{{\mathcal {R}}} \,:\, | {\mathcal {R}} | = k \}$ of indicator statistics corresponding to rook placements ${\mathcal {R}} \subseteq [n] \times [n]$ of size k is a spanning set of ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}})$ , but this spanning set is almost always linearly dependent. In [Reference Hamaker and Rhoades10, Cor. 4.7] it is proven that when ${\mathbb {F}} = \mathbb {R}$ is the field of real numbers, the dimension of ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}})$ equals to the number $a_{n,n-k} + \cdots + a_{n,n-1} + a_{n,n}$ of permutations in ${\mathfrak {S}}_n$ which have an increasing subsequence of length at least $n-k$ . The methods of [Reference Hamaker and Rhoades10] apply whenever ${\mathbb {F}}$ has characteristic 0 or characteristic $p> n$ ; we will see (Theorem 3.16) that this is true over any field.

The paper [Reference Hamaker and Rhoades10] did not give an explicit basis of the space of k-local statistics consisting of statistics of the form ${\mathbf {1}}_{\mathcal {R}}$ ; we solve this problem in Theorem 3.16 below. Although the members ${\mathbf {1}}_{\mathcal {R}}$ of our basis for ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}})$ can correspond to rook placements with $|{\mathcal {R}}| < k$ in general, we will obtain a nested family of bases for the chain of vector spaces ${\mathrm {Loc}}_0({\mathfrak {S}}_n,{\mathbb {F}}) \subseteq {\mathrm {Loc}}_1({\mathfrak {S}}_n,{\mathbb {F}}) \subseteq \cdots \subseteq {\mathrm {Loc}}_{n-1}({\mathfrak {S}}_n,{\mathbb {F}})$ . To achieve these goals, we recall a standard fact about associated graded ideals.

Let ${\mathbf {x}}$ be a finite set of variables, and consider the polynomial ring ${\mathbb {F}}[{\mathbf {x}}]$ over these variables. Given $d \geq 0$ and a graded ${\mathbb {F}}$ -algebra A, let $A_{\leq d} \subseteq A$ be the subspace of elements of degree at most d. We have a filtration ${\mathbb {F}}[{\mathbf {x}}]_{\leq 0} \subseteq {\mathbb {F}}[{\mathbf {x}}]_{\leq 1} \subseteq {\mathbb {F}}[{\mathbf {x}}]_{\leq 2} \subseteq \cdots $ of ${\mathbb {F}}[{\mathbf {x}}]$ by finite-dimensional subspaces.

Lemma 3.15 is the heart of the orbit harmonics isomorphisms (2.7). We include its straightforward proof for completeness.

Proof. If ${\mathcal {B}}$ were not linearly independent modulo $I \cap {\mathbb {F}}[{\mathbf {x}}]_{\leq d}$ , there would exist scalars $c_b \in {\mathbb {F}}$ not all zero and an element $g \in I$ with $\deg (g) \leq d$ such that $\sum _{b \in {\mathcal {B}}} c_b \cdot b = g$ . Since the elements of ${\mathcal {B}}$ are homogeneous, taking the highest degree component of both sides of this equation would result in a linear dependence of ${\mathcal {B}}$ modulo ${\mathrm {gr}} \, I$ , a contradiction.

If ${\mathcal {B}}$ did not span ${\mathbb {F}}[{\mathbf {x}}]_{\leq d} / (I \cap {\mathbb {F}}[{\mathbf {x}}]_{\leq d} )$ , there would be some homogeneous polynomial $h \in {\mathbb {F}}[{\mathbf {x}}]_{\leq d}$ such that g does not lie in the span of ${\mathcal {B}}$ modulo $I \cap {\mathbb {F}}[{\mathbf {x}}]_{\leq d}$ . Choose such an h with $\deg (h)$ minimal. There exist scalars $c_b \in {\mathbb {F}}$ such that $\sum _{b \in {\mathcal {B}}} c_b \cdot b = h + \tau (g)$ for some $g \in I$ with $\deg (g) = \deg (h)$ (so that in particular $g \in I \cap {\mathbb {F}}[{\mathbf {x}}]_{\leq d}$ ), where $\tau (g)$ is the highest degree component of g. Discarding redundant terms if necessary, we may assume that $c_b = 0$ whenever $\deg (b) \neq \deg (h)$ . We conclude that $h + g - \sum _{b \in {\mathcal {B}}} c_b \cdot b$ has degree $< \deg (h)$ , so by our choice of h there exist $c^{\prime }_b \in {\mathbb {F}}$ and $g' \in I \cap {\mathbb {F}}[{\mathbf {x}}]_{\leq d}$ with

$$ \begin{align*} \sum_{b \in {\mathcal {B}}} c^{\prime}_b \cdot b = h + g - \sum_{b \in {\mathcal {B}}} c_b \cdot b + g' \end{align*} $$

so that $h = \sum _{b \in {\mathcal {B}}} (c^{\prime }_b - c_b) \cdot b - (g' + g)$ lies in the span of ${\mathcal {B}}$ modulo $I \cap {\mathbb {F}}[{\mathbf {x}}]_{\leq d}$ , a contradiction.

An application of Lemma 3.15 gives a basis of the vector space ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}})$ .

Theorem 3.16. The vector space ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}})$ of k-local statistics ${\mathfrak {S}}_n \rightarrow {\mathbb {F}}$ has basis

(3.29) $$ \begin{align} \{ {\mathbf {1}}_{{\mathcal {S}}(w)} \,:\, w \in {\mathfrak {S}}_n, \, \, {\mathrm {lis}}(w) \geq n-k \} \end{align} $$

given by indicator functions of shadow sets of permutations in ${\mathfrak {S}}_n$ which contain an increasing subsequence of length $n-k$ .

Other authors (see, e.g., [Reference Dafni, Filmus, Lifshitz, Lindzey and Vinyals5]) refer to the functions ${\mathbf {1}}_{\mathcal {R}}$ as juntas. So Theorem 3.16 describes a basis of shadow juntas.

Proof. For $\ell \leq k$ , any $\ell $ -local permutation statistic is also k-local, so the indicator functions in question are members of ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}})$ by Lemma 3.5. Identifying ${\mathfrak {S}}_n = P_n$ with the locus of permutation matrices in ${\mathbb {F}}^{n \times n}$ , the indicator function ${\mathbf {1}}_{{\mathcal {R}}}$ corresponding to a rook placement ${\mathcal {R}} \subseteq [n] \times [n]$ is represented by the degree $|{\mathcal {R}}|$ monomial $m({\mathcal {R}}) \in {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ . It follows that we have an isomorphism

(3.30) $$ \begin{align} {\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}}) \cong {\mathbb {F}}[{\mathbf {x}}_{n \times n}]_{\leq k} / ( {\mathbf {I}}(P_n) \cap {\mathbb {F}}[{\mathbf {x}}_{n \times n}]_{\leq k} ) \end{align} $$

of ${\mathbb {F}}$ -vector spaces given by ${\mathbf {1}}_{\mathcal {R}} \mapsto m({\mathcal {R}}) + ( {\mathbf {I}}(P_n) \cap {\mathbb {F}}[{\mathbf {x}}_{n \times n}]_{\leq k} )$ . Write

(3.31) $$ \begin{align} {\mathcal {B}} = \{ {\mathfrak {s}}(w) \,:\, w \in {\mathfrak {S}}_n \text{ has an increasing subsequence of length at least}\ n-k \} \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}] \end{align} $$

for the set of monomials representing the indicator functions in the statement. Theorem 3.12 implies that ${\mathcal {B}}$ descends to a basis for the ${\mathbb {F}}$ -vector space $( {\mathbb {F}}[{\mathbf {x}}_{n \times n}] / {\mathrm {gr}} \, {\mathbf {I}}(P_n) )_{\leq k}$ . An application of Lemma 3.15 shows that ${\mathcal {B}}$ also descends to a basis for ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]_{\leq k} / ( {\mathbf {I}}(P_n) \cap {\mathbb {F}}[{\mathbf {x}}_{n \times n}]_{\leq k} )$ , and the isomorphism (3.30) completes the proof.

The nested shadow junta bases of ${\mathrm {Loc}}_0({\mathfrak {S}}_3,{\mathbb {F}}) \subset {\mathrm {Loc}}_1({\mathfrak {S}}_3,{\mathbb {F}}) \subset {\mathrm {Loc}}_2({\mathfrak {S}}_3,{\mathbb {F}})$ are as follows.

It may be interesting to find a basis of ${\mathrm {Loc}}_k({\mathfrak {S}}_n,{\mathbb {F}})$ drawn from the spanning set $\{ {\mathbf {1}}_{\mathcal {R}} \,:\, |{\mathcal {R}}| = k \}$ . By Theorem 3.12, the above monomials also form a vector space basis of ${\mathbb {F}}[{\mathbf {x}}_{3 \times 3}]/I_3$ .

The results we have proven so far hold when the field ${\mathbb {F}}$ is replaced by a commutative ring R. More precisely, we have an ideal $I_n^R \subseteq R[{\mathbf {x}}_{n \times n}]$ with the same generating set as in Definition 1.1.

• • The proofs of Lemmas 3.1 and 3.11 goes through to show that the shadow monomials $\{ {\mathfrak {s}}(w) \,:\, w \in {\mathfrak {S}}_n \}$ span $R[{\mathbf {x}}_{n \times n}]/I_n^R$ over R. Here, we use the fact that the coefficients in the polynomials $a_{S,T}, b_{S,T}$ appearing in Lemma 3.1 are all $\pm 1$ .

• • When $R = {\mathbb {Z}}$ , a linear dependence of $\{ {\mathfrak {s}}(w) \,:\, w \in {\mathfrak {S}}_n \}$ modulo $I_n^{\mathbb {Z}}$ would induce a linear dependence modulo $I_n^{\mathbb {Q}}$ . By Theorem 3.12, $\{ {\mathfrak {s}}(w) \,:\, w \in {\mathfrak {S}}_n \}$ descends to a ${\mathbb {Z}}$ -basis of ${\mathbb {Z}}[{\mathbf {x}}_{n \times n}]/I_n^{\mathbb {Z}}$ .

• • Since $R[{\mathbf {x}}_{n \times n}]/I_n^R = R \otimes _{\mathbb {Z}} {\mathbb {Z}}[{\mathbf {x}}_{n \times n}]/I_n^{\mathbb {Z}}$ , the set $\{ {\mathfrak {s}}(w) \,:\, w \in {\mathfrak {S}}_n \}$ descends to a R-basis of $R[{\mathbf {x}}_{n \times n}]/I_n^R$ for any R. The proof of Lemma 3.15 holds over R, so the shadow juntas $\{ {\mathbf {1}}_{{\mathcal {S}}(w)} \,:\, w \in {\mathfrak {S}}_n, \, \, {\mathrm {lis}}(w) \geq n-k \}$ form an R-basis of ${\mathrm {Loc}}_k({\mathfrak {S}}_n,R)$ .