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#Kohnert polynomials

Kohnert polynomials were introduced by S. Assaf and D. Searles in [AS22]. This is a large family of polynomials, which contains several of the classical families as special cases, in particular the family of Schubert polynomials and key polynomials.

All Kohnert polynomials are monomial slide positive [Thm. 3.7, AS22], and they were later proved to be key-positive in [Ass21].

See [AABE23] for a representation-theoretical interpretation of Kohnert polynomials through flagged Schur modules.

#Definition

We use French coordinates for diagrams: rows are indexed from bottom to top, starting with row \(1.\) A diagram is a finite set of cells in \(\setN \times \setN.\) Its weight is the weak composition \[\operatorname{wt}(D)_i \coloneqq \#\{ \text{cells of } D \text{ in row } i\}.\]

Definition (Kohnert move).

A Kohnert move selects the rightmost cell in a row and moves it downward in its column to the first empty position below it, jumping over occupied cells if necessary. Let \(\mathrm{KD}(D)\) be the set of diagrams obtained from \(D\) by a sequence of Kohnert moves.

Definition

The Kohnert polynomial of a diagram \(D\) is \[\kohnert_D(\xvec) \coloneqq \sum_{T \in \mathrm{KD}(D)} \xvec^{\operatorname{wt}(T)}.\]

In our French coordinate convention, a diagram \(D\) is southwest if, whenever \(D\) contains cells \((c_1,r_2)\) and \((c_2,r_1)\) with \(c_1\lt c_2\) and \(r_1\lt r_2,\) it also contains \((c_1,r_1).\) This is the coordinate-reflected version of the northwest diagrams used in [AABE23].

For a weak composition \(a=(a_1,\dotsc,a_n),\) the key diagram \(D(a)\) is the left-justified diagram with \(a_i\) cells in row \(i.\) For a permutation \(w,\) let \(D(w)\) denote its Rothe diagram.

Theorem (Kohnert, Winkel, Assaf–Searles [Thms. 2.6 and 2.10, AS22]).

For every weak composition \(a,\) \[\kohnert_{D(a)}(\xvec)=\key_a(\xvec).\] For every permutation \(w,\) \[\kohnert_{D(w)}(\xvec)=\schubert_w(\xvec).\]

That is, we can produce both key polynomials and Schubert polynomials via Kohnert diagrams. S. Assaf gives an explicit bijection between Kohnert diagrams for Rothe diagrams and Billey–Jockusch–Stanley compatible sequences [Ass22]. This gives a bijective proof of Kohnert’s rule for Schubert polynomials.

Theorem (Assaf; Armon–Assaf–Bowling–Ehrhard).

If \(D\) is southwest, then \(\kohnert_D(\xvec)\) is key-positive. Moreover, in the northwest convention of [AABE23], the characters of flagged Schur modules are computed by Kohnert’s rule exactly for northwest indexing diagrams.

Thus Kohnert’s rule covers the module characters whose special cases include flagged skew Schur polynomials, Demazure characters, and Schubert polynomials. For arbitrary skew box arrangements, the southwest/northwest closure condition is the useful test: without it, the simple Kohnert-crystal and divided-difference recurrences need not survive unchanged.

Example (A first Kohnert computation).

Let \(a=(0,2).\) The key diagram \(D(a)\) has two cells in row \(2.\) There are three Kohnert diagrams: \[\begin{array}{c|cc} 2 & \bullet & \bullet \\ 1 & & \\ \hline & 1 & 2 \end{array} \qquad \begin{array}{c|cc} 2 & \bullet & \\ 1 & & \bullet \\ \hline & 1 & 2 \end{array} \qquad \begin{array}{c|cc} 2 & & \\ 1 & \bullet & \bullet \\ \hline & 1 & 2 \end{array}.\] Their weights are \((0,2),\) \((1,1),\) and \((2,0),\) respectively. Therefore \[\kohnert_{D(0,2)}(x_1,x_2) = x_2^2+x_1x_2+x_1^2 = \key_{(0,2)}(x_1,x_2).\] This is the smallest example where a Kohnert move produces more than the initial monomial.

#Kohnert quasisymmetric functions

The Kohnert quasisymmetric function of a diagram \(D\) is the stable limit \[\kohnert_D(\xvec) \coloneqq \lim_{m\to\infty} \kohnert_{0^m\times D}(\xvec),\] where \(0^m\times D\) is obtained from \(D\) by shifting all cells upward by \(m\) rows. These stable limits are well-defined and expand positively in the monomial quasisymmetric basis and in the fundamental quasisymmetric basis [AS22]. For lock diagrams, the stable limits are the extended Schur functions.

Example (A stable key limit).

Let \(D=D(1,1,2)\) be the left-justified key diagram with one cell in each of the first two rows and two cells in the third row. The stable Kohnert quasisymmetric function is the stable limit of the corresponding key polynomials, so in this case \[\kohnert_D(\xvec) = \schurS_{211}(\xvec).\] In the fundamental quasisymmetric basis this is \[\kohnert_D(\xvec) = \gessel_{112}+\gessel_{121}+\gessel_{211},\] and in the monomial quasisymmetric basis it is \[\kohnert_D(\xvec) = 3\qmonom_{1111} +\qmonom_{112} +\qmonom_{121} +\qmonom_{211}.\] This gives a small concrete bridge between Kohnert diagrams, key polynomials, and the Schur functions obtained after stabilization.

#Kohnert tableaux

Kohnert tableaux were introduced by S. Assaf and D. Searles in [AS18]. They are a labeled version of Kohnert diagrams for key diagrams. The labels record which row of the initial key diagram a cell came from.

Definition (Kohnert tableau [Def. 2.3, AS18]).

Let \(a=(a_1,\dotsc,a_n)\) be a weak composition. A Kohnert tableau of content \(a\) is a filled diagram using entries \[1^{a_1},2^{a_2},\dotsc,n^{a_n},\] one entry per cell, satisfying the following conditions.

  1. There is exactly one \(i\) in each column \(1,\dotsc,a_i.\)

  2. If a cell with entry \(i\) lies in row \(r,\) then \(i\geq r.\)

  3. The cells with entry \(i\) weakly descend from left to right.

  4. If \(i\lt{}j\) appear in the same column with \(i\) above \(j,\) then there is an \(i\) in the column immediately to the right and strictly above \(j.\)

Let \(\mathrm{KT}(a)\) be the set of Kohnert tableaux of content \(a.\) The weight \(\operatorname{wt}(T)\) is the weak composition counting cells of \(T\) by row.

Theorem (Assaf–Searles [Thm. 2.8, AS18]).

There is a weight-preserving bijection between Kohnert diagrams in \(\mathrm{KD}(D(a))\) and Kohnert tableaux in \(\mathrm{KT}(a).\) Hence \[\key_a(\xvec) = \sum_{T \in \mathrm{KT}(a)} \xvec^{\operatorname{wt}(T)}.\]

Definition (Quasi-Yamanouchi Kohnert tableau [Def. 2.10, AS18]).

A Kohnert tableau is quasi-Yamanouchi if, for each nonempty row \(i,\) either row \(i\) contains an entry \(i,\) or row \(i+1\) contains a cell weakly to the right of a cell in row \(i.\) Let \(\mathrm{QKT}(a)\) denote the set of quasi-Yamanouchi Kohnert tableaux of content \(a.\)

The quasi-Yamanouchi condition gives the compact expansion of key polynomials in the fundamental slide basis.

Theorem (Assaf–Searles [Thm. 2.13, AS18]).

For every weak composition \(a,\) \[\key_a(\xvec) = \sum_{T \in \mathrm{QKT}(a)} \slideF_{\operatorname{wt}(T)}(\xvec).\]

#Quasi-key polynomials

S. Assaf and D. Searles also use Kohnert tableaux to define quasi-key polynomials, which lift quasi-Schur functions from the quasisymmetric setting to the full polynomial ring.

Definition (Quasi-key polynomial [Def. 3.2, AS18]).

A quasi-Kohnert tableau is a Kohnert tableau satisfying two additional conditions from [Def. 3.1, AS18]. Let \(\mathrm{qKT}(a)\) be the set of quasi-Kohnert tableaux of content \(a.\) The quasi-key polynomial indexed by \(a\) is \[Q_a(\xvec) \coloneqq \sum_{T \in \mathrm{qKT}(a)} \xvec^{\operatorname{wt}(T)}.\]

Theorem (Assaf–Searles [Thms. 3.3, 3.4, and 3.7, AS18]).

The quasi-key polynomials \(Q_a\) form a \(\setZ\)-basis of the polynomial ring. They have a positive fundamental slide expansion \[Q_a(\xvec) = \sum_{T \in \mathrm{QqKT}(a)} \slideF_{\operatorname{wt}(T)}(\xvec),\] where the sum is over quasi-Yamanouchi quasi-Kohnert tableaux. Moreover, each key polynomial expands positively in the quasi-key basis: \[\key_a(\xvec) = \sum_{b\in \mathrm{Qlswap}(a)} Q_b(\xvec).\]

Theorem (Assaf–Searles [Thms. 4.15 and 4.16, AS18]).

If the nonzero entries of \(a\) occur in an interval ending in position \(k,\) then \[Q_a(x_1,\dotsc,x_k) = \mathrm{QS}_{\operatorname{flat}(a)}(x_1,\dotsc,x_k),\] where \(\mathrm{QS}_{\alpha}\) is the quasi-Schur function. In general, \[\lim_{m\to\infty} Q_{0^m\times a}(X) = \mathrm{QS}_{\operatorname{flat}(a)}(X).\]

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