The symmetric functions catalog

An overview of symmetric functions and related topics


Key polynomials

The key polynomials, also known as Demazure characters, is a family of non-symmetric polynomials, indexed by weak compositions, or alternatively, an integer partition and a permutation. The set $\{ \key_\alpha(x_1,\dotsc,x_n) \}_\alpha$ where $\alpha$ ranges over all compositions of length $n,$ is a basis for $\setC[x_1,\dotsc,x_n].$

The key polynomials generalize the Schur polynomials.

Demazure characters were introduced in [Dem74bDem74a]. Later, a combinatorial formula using so called key tableaux was discovered in [LS90a]. Key polynomials are also given as a specialization of non-symmetric Macdonald polynomials, see [HHL08], and this gives several combinatorial models for the key polynomials, see the works of S. Mason [Mas09], and also [Kur16AS19a].

Key polynomials are closely related to Demazure atoms.

The K-theoretic version of key polynomials are the Lascoux polynomials.

Definition (Operators)

Define the divided difference operator $\partial_i$ as

\[ \partial_i(f) = \frac{f - s_i(f)}{z_i - z_{i+1}}, \]

and let $\pi_i(f) \coloneqq \partial_i(z_i f)$ for $i=1,\dotsc,n-1$ whenever $f \in \setC[z_1,\dotsc,z_n].$ One can show that $\pi_i^2 = \pi_i$ and that they satisfiy the braid relations.

Let $\sigma = s_{i_1} s_{i_2} \dotsc s_{i_\ell}$ be a reduced word of a permutation $\sigma \in \symS_n,$ and let

\[ \pi_\sigma \coloneqq \pi_{i_1} \circ \pi_{i_2} \circ \dotsb \circ \pi_{i_\ell}. \]

This is well-defined, as the action of $\pi_\sigma$ is independent of the choice of reduced word.

Let $\lambda$ be a partition with at most $n$ parts, and let $\sigma \in \symS_n.$ The key polynomial $\key_{\lambda,\sigma}(z)$ is defined as

\begin{equation*} \key_{\lambda,\sigma}(z) \coloneqq \pi_\sigma\left( z_1^{\lambda_1} \dotsm z_n^{\lambda_n} \right). \end{equation*}

This definition shows the close connection between Schubert polynomials and key polynomials. Notice that if we take $w_0$ to be the longest permutation in $\symS_n,$ then

\[ \schurS_\lambda(z_1,\dotsc,z_n) = \key_{\lambda,w_0}(z_1,\dotsc,z_n), \]

so the key polynomials are a superfamily of the Schur polynomials.

Example (Taken from [AA19a]).

Let $\lambda = (2,1,0,0)$ and $\sigma = [2,4,3,1] \in \symS_4$ in one-line notation. The permutation can be expressed as a reduced word as $\sigma = s_2 s_3 s_2 s_1.$ We compute the key polynomial as follows:

\begin{align*} \key_{\lambda,\sigma}(z) &= \pi_2 \pi_3 \pi_2 \pi_1 (z_1^2 z_2) = \pi_2 \pi_3 \pi_2 \partial_1 (z_1^3 z_2 ) = \pi_2 \pi_3 \pi_2 \left( \frac{z_1^3z_2 - z_1 z^3_2}{z_1-z_2}\right) \\ &= \pi_2 \pi_3 \pi_2 (z_1^2z_2 + z_1 z_2^2). \end{align*}

We continue the calculation by applying $\pi_2$ and get

\begin{align*} \key_{\lambda,\sigma}(z) &= \pi_2 \pi_3 (z_2 z_1^2+z_3 z_1^2+z_2^2 z_1+z_3^2 z_1+z_2 z_3 z_1). \end{align*}

Applying $\pi_2 \pi_3$ then finally gives

\begin{equation*}\label{eq:keyExample} \begin{split} \key_{\lambda,\sigma}(z) &= z_1^2 z_2 + z_1^2 z_3 + z_1^2 z_4 + z_1 z_2^2 + z_1 z_3^2 \\ &\phantom{=}+ z_1 z_4^2 + z_1 z_2 z_3 + z_1 z_2 z_4 + z_1 z_3 z_4 . \end{split} \end{equation*}

In general, some monomials may appear multiple times.

Definition (GT-patterns)

A formula for the key polynomials as a sum over lattice points in a union of faces of Gelfand–Tsetlin polytopes was proved in [KST12].

Theorem (See [KST12]).

Let $\GT(\lambda,\sigma)$ be defined as the polytopal complex

\[ \GT(\lambda,\sigma) \coloneqq \bigcup_{\substack{\mathcal{F} \in \GT(\lambda) \\ \type(\mathcal{F}) = w_0\sigma}} \mathcal{F}. \]

That is, $\GT(\lambda,\sigma)$ is the union of all reduced Kogan faces of type $w_0\sigma$ in the polytope $\GT(\lambda).$

The key polynomial $\key_{\lambda,\sigma}(z)$ can be computed as

\begin{align}\label{eq:keyGTFormula} \key_{\lambda,\sigma}(z_1,\dotsc,z_n) = \sum_{G \in \GT(\lambda,\sigma) \cap \setZ^{\tfrac{n(n+1)}{2}} } z_1^{w_1(G)} \dotsm z_n^{w_n(G)} \end{align}

Fix $\lambda,$ $\sigma \in \symS_n.$ With the above theorem, it follows that for the map

\[ k \mapsto \key_{k \lambda,\sigma}(1^n) \]

is a polynomial in $k.$ This is in fact an Ehrhart polynomial of a union of faces in a GT-polytope. Let $P_{\sigma}(\lambda_1,\dotsc,\lambda_n;k)$ denote this polynomial. From the definition via divided difference operators, one can see that $P_{\sigma}(\lambda_1,\dotsc,\lambda_n;k)$ is a polynomial in $\setQ[\lambda_1,\dotsc,\lambda_n,k].$

See [AA19a] for the following conjecture.

Conjecture (Alexandersson–Alhajjar (2018)).

The polynomial $P_{\sigma}(\lambda_1,\dotsc,\lambda_n;k)$ has non-negative coefficients.

Moreover, after setting $a_j = \lambda_1+\lambda_2+\dotsb+\lambda_j,$ the function $P_{\sigma}(a_1,\dotsc,a_n;k)$ is a polynomial in $\setQ[a_1,\dotsc,a_n,k]$ with non-negative coefficients.

See also my conjecture regarding the corresponding $h^*$-polynomial.

Definition (Skyline fillings)

Definition (SSYT)

Representation theory of GLn

See introduction in e.g.

Products of key polynomials

A product of key polynomials is not in general key positive. However, in some instances such a product is key positive, see [Kou18].

Pieri rule for key polynomials

A Pieri rule for key polynomials is described via an Robinson–Schensted–Knuth-type algorithm is given by S. Assaf and D. Quijada in

Skew key polynomials

In AssafvanWilligenburg2019, the authors introduce skew key polynomials, and show that these are positive in the key basis. They use weak dual equivalence to obtain their result. This generalizes an earlier result by Reiner and Shimozono, [RS95].

Demazure atoms

The Demazure atoms (originally called standard bases) may be defined in a manner similar to key polynomials using operators. They were introduced by Lascoux and Schützenberger in [LS90a]. Using the notation for divided difference operators, let $\theta_i \coloneqq \pi_i -1.$ These satisfy the braid relations and we have $\theta_i \theta_j = \theta_j \theta_i$ whenever $|i-j| \geq 2.$

Let $\lambda$ be a partition with at most $n$ parts, and let $\sigma \in \symS_n.$ The Demazure atom $\atom_{\lambda,\sigma}(z)$ is defined as

\begin{equation*} \atom_{\lambda,\sigma}(z) \coloneqq \theta_\sigma\left( z_1^{\lambda_1} \dotsm z_n^{\lambda_n} \right). \end{equation*}

Since $\pi_i = \theta_i+1,$ one can easily show that key polynomials are positive in the Demazure atom basis. In fact we have that

\[ \key_{\lambda,\sigma}(z) = \sum_{\tau \leq \sigma} \atom_{\lambda,\tau}(z). \]

Here, $\tau \leq \sigma$ is the Bruhat order.

See [Pun16] for a good overview of Demazure atoms, key polynomials and recent results.

Cauchy kernel

A bijective proof of the Cauchy-type identity below is given in [Thm. 6, Las03]. The proof uses the Robinson–Schensted–Knuth correspondence.

Theorem (See [Thm. 6, Las03]).

We have that

\[ \prod_{i+j \leq n+1} (1-x_iy_j)^{-1} = \sum_{\alpha \in \setN^n} \atom_{w_0\alpha}(\xvec) \key_{\alpha}(\yvec). \]

In [FL09], the authors prove this formula and several other Cauchy-type identities involving key polynomials and Demazure atoms in other types.

A generalization to truncated staircase shapes is provided by O. Azenhas and A. Emami in [AE15]. They use a version of RSK introduced by S. Mason in [Mas08].

An alternative proof also appear in [CK17] where the authors use $U_q(\mathfrak{g})$ crystals.

Example (Personal communication, Vasu Tewari 2019).

Recall the Cauchy identity for Schubert polynomials (see e.g. [p. 30, PS08]).

\[ \sum_{w \in \symS_n} \schubert_w(\xvec) \schubert_{w w_0}(\yvec) = \prod_{i+j \leq n} (1 + x_iy_j) = \prod_{k=1}^{n-1}\sum_{j=0}^k y_{n-k}^{k-j} \elementaryE_{j}(x_1,\dotsc,x_k). \]

Since the Schubert polynomials are positive in the key basis, it follows that the standard elementary monomials are key-positive.

Demazure keys and Demazure atoms in type $B,$ $C$ and $D$

In [FL09], A. Fu and A. Lascoux define divided difference operators that allow us to define key polynomials and Demazure atoms in types $B,$ $C$ and $D.$

See these slides for a nice overview. In particular, the type $C$ key polynomials contain the symplectic Schur functions. There is an operator definition, crystal graph definition and a key-tableau definition of these.

They end with an open question about how to characterize type $B$ Demazure atoms as a sum over fillings. This question is perhaps answered in the recent preprint

See also [RY11], as key polynomials are specializations of non-symmetric Macdonald polynomials.

Type $B/C$ key polynomials

Type $B/C$ Demazure atoms

See [FL09] for definitions of key polynomials in other types. These are related to the orthogonal and symplectic Schur functions.