The symmetric functions catalog

#Key polynomials

The key polynomials, also known as Demazure characters, form a family of nonsymmetric polynomials, indexed by weak compositions, or alternatively, by an integer partition and a permutation. The set \(\{ \key_\alpha(x_1,\dotsc,x_n) \}_\alpha,\) where \(\alpha\) ranges over all weak 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 [Dem74, Dem74]. Geometrically, they are characters of Demazure modules associated with Schubert varieties in the flag variety; see varieties in algebraic combinatorics. Later, a combinatorial formula using so-called key tableaux was discovered in [LS90]. Key polynomials are also given as a specialization of nonsymmetric Macdonald polynomials; see [HHL08]. This gives several combinatorial models for the key polynomials; see the work of S. Mason [Mas09], and also [Kur16, AS19]. Another positive model is given by Kohnert tableaux, which label the Kohnert diagrams for the key diagram. The same Kohnert-tableau framework gives positive expansions into the quasi-key basis and into the fundamental slide basis [AS18].

Y. Xiao, R. Xiong, and H. Zhang introduce hybrid pipe dreams for key polynomials: a family of tile models linked by explicit local weight-preserving bijections, with a two-sided branching rule as an application [XXZ26]. Their construction is modeled on the hybrid approach for Schubert polynomials and complements the classical pipe-dream formula for Schubert polynomials.

Mrigendra Singh Kushwaha, K. N. Raghavan, and S. Viswanath give simple procedures for computing the left and right keys of a semistandard Young tableau [KRV23]. Their procedures repeatedly apply Deodhar lifts, which recover the final and initial directions of the corresponding Littelmann paths.

L. Lapointe and L. Pena give a non-algorithmic characterization particularly well adapted to RSK [LP26].

This makes the right key visibly compatible with Knuth equivalence and the RSK correspondence. Lapointe and Pena use it to construct a tableau generating function for the \(m\)-symmetric Schur functions at \(t=0\), identify the dual family with key polynomials, and prove the corresponding Cauchy identity. Their specialization also gives an ordinary-RSK proof of A. Lascoux’s nonsymmetric Cauchy identity for Demazure characters and atoms.

Key polynomials are closely related to Demazure atoms.

The \(K\)-theoretic versions of key polynomials are the Lascoux polynomials.

#Almost symmetric Schur functions

M. B. Weising introduces the almost symmetric Schur functions \(s_{(\mu|\lambda)}\) [Wei24]. They are the \(q=t=0\) specialization of stable-limit nonsymmetric Macdonald functions and simultaneously generalize finite-variable key polynomials and ordinary Schur functions. The paper gives recurrence definitions using isobaric divided differences and Weyl symmetrization operators, a tableau formula, and positive expansions in the monomial and monomial-Schur bases for almost symmetric functions.

#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 the isobaric divided difference operator be \(\pi_i(f) \coloneqq \partial_i(z_i f)\) for \(i=1,\dotsc,n-1\) whenever \(f \in \setC[z_1,\dotsc,z_n].\) The operators satisfy \(\pi_i^2 = \pi_i\) and 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 \(\sigma \in \symS_n.\) In fact, we can choose \(\sigma\) to be the minimal coset representation for the parabolic subgroup \(W_J \subseteq \symS_n\) that fixes \(\lambda.\) The key polynomial \(\key_{\lambda,\sigma}(z)\) is defined as \[\key_{\lambda,\sigma}(z) \coloneqq \pi_\sigma\left( z_1^{\lambda_1} \dotsm z_n^{\lambda_n} \right).\] Note that if \(\sigma^1(\lambda)=\sigma^2(\lambda),\) then \(\key_{\lambda,\sigma^1}(z) = \key_{\lambda,\sigma^2}(z).\)

This definition shows the close connection between Schubert polynomials and key polynomials. If \(w_0\) is the longest permutation in \(\symS_n,\) then \[\schurS_\lambda(z_1,\dotsc,z_n) = \key_{\lambda,w_0}(z_1,\dotsc,z_n),\] so key polynomials generalize the Schur polynomials.

Alternatively, key polynomials are also commonly indexed by weak compositions. If \(\alpha = (\alpha_1,\dotsc,\alpha_n)\) is a weak composition, we let \(\key_{\alpha}(z)\) be the key polynomial \(\key_{\lambda,\sigma}(z),\) where \(\lambda\) is the partition obtained from \(\alpha\) by sorting the elements in decreasing order, and \(\sigma \in \symS_n\) is a permutation such that it sorts \(\alpha\) into a partition; \(\lambda = (\alpha_{\sigma(1)},\alpha_{\sigma(2)},\dotsc,\alpha_{\sigma(n)}).\)

For example, \(\alpha = (1,0,3,2)\) corresponds to \(\lambda = (3,2,1,0)\) and \(\sigma = [3, 4, 1, 2].\) With these conventions, \(\key_{\lambda}(z)=\schurS_{\lambda}(z).\)

Note: This indexing convention is not uniform in the literature; in some places, one needs to reverse the indexing composition \(\alpha.\)

#Kogan faces and key polynomials via GT-polytopes

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]. This subsection describes the construction in detail, following [KST12] and [Kog00].

Recall that the GT-polytope \(\GT(\lambda)\) is defined by the interlacing inequalities \(x_{i+1,j} \geq x_{i,j} \geq x_{i+1,j+1}\) together with the boundary condition \(x_{n,j} = \lambda_j.\) Its integer points are in bijection with semistandard Young tableaux of shape \(\lambda.\) The weight of a GT-pattern \(G\) is \(w(G) = (w_1, \dotsc, w_n)\) where \(w_k = \sum_j x_{k,j} - \sum_j x_{k-1,j}\) (row sum differences, with \(\sum_j x_{0,j} = 0\)).

#Key polynomials from Kogan faces

The key polynomial \(\key_{\lambda,\sigma}\) is indexed by a partition \(\lambda\) and a permutation \(\sigma \in \symS_n.\) The connection to Kogan faces uses the product \(w_0 \sigma,\) where we use function composition: \((w_0 \sigma)(i) = w_0(\sigma(i))\); that is, \(\sigma\) acts first.

V. Kiritchenko constructs simple geometric mitosis operations on faces of Cayley sums of polytopes [Kir25]. These operations act as convex-geometric counterparts of divided difference operators and give a uniform construction of Knutson–Miller mitosis in type \(A\) and Fujita mitosis in type \(C\) on Kogan faces.

#Kempf permutations and flagged Schur polynomials

#Ehrhart polynomial

Fix \(\lambda,\) \(\sigma \in \symS_n.\) With the above theorem, it follows that 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 [AA19] for the following conjecture.

See also the related conjecture regarding the corresponding \(h^*\)-polynomial.

#Definition (Skyline fillings)

In [HHL08], the authors presented a combinatorial model for the nonsymmetric Macdonald polynomials. Setting \(q=t=0\) in the nonsymmetric Macdonald polynomial \(\macdonaldE_{\alpha}(\xvec,q,t)\) recovers the key polynomial \(\key_\alpha(\xvec).\)

From the work of S. Mason [Mas09], we in fact get two combinatorial models for key polynomials as special cases of the permuted basement Macdonald polynomials. In the definition below, we assume that the reader is familiar with the combinatorial model for the permuted basement Macdonald polynomials.

A series of bijections proves the equivalence of the two above formulas; see [AS19, Prop. 5.5] (with \(t=0\)). This amounts to proving that interchanging two adjacent rows (including the basement) in an augmented diagram in such a way that a longer row is moved upwards, preserves the weighted sum of SSAFs with corresponding shape and basement.

For classical background on tableaux, Gelfand–Tsetlin patterns, and branching rules around key-tableau combinatorics, see [Pro94]. The Bruhat-order side of the divided-difference formulas and the standard elementary monomials below is closely related to the chain constructions of A. Postnikov and R. P. Stanley [PS08].

#Definition (SSYT)

A key tableau is a semistandard tableau such that the entries in column \(j+1\) are a subset of the entries in column \(j,\) for all \(j.\)

Below is the unique key tableau with weight \((1,3,0,1,2,4)\):

The map that sends a key to its weight is a bijection between the set of key tableaux and the set of compositions. Given a composition \(\alpha = (\alpha_1,\dotsc,\alpha_\ell),\) place a cell \(j\) in the first \(\alpha_j\) columns. This produces a unique key tableau, \(\mathrm{key}(\alpha),\) of shape \(\sort(\alpha).\)

R. Hodges and A. Yong characterize multiplicity-free key polynomials [HY22]. They also relate Coxeter combinatorics to spherical Schubert geometry [HY20].

Given a semistandard tableau \(T,\) we can compute its right key \(K_+(T),\) via a direct but involved process. A simpler method is proved in [Mas09]. The key polynomial can then be computed by \[\key_\alpha(\xvec) = \sum_{\substack{T \in \SSYT(\sort(\alpha)) \\ K_+(T) \leq \mathrm{key}(\alpha)}} \xvec^T\] where we sum over all semistandard tableaux \(T\) whose right key, \(K_+(T),\) is entrywise weakly smaller than \(\mathrm{key}(\alpha).\)

Similarly, the Demazure atoms are given as the sum over semistandard tableaux with a given right key: \[\atom_\alpha(\xvec) = \sum_{\substack{T \in \SSYT(\sort(\alpha)) \\ K_+(T) = \mathrm{key}(\alpha)}} \xvec^T\]

#Keys via Kostant partition function

Let \(\Phi^+=\{ \evec_i-\evec_j : 1 \leq i \lt{} j \leq n\}\) be the positive roots of the root system of type \(A_{n-1}.\) For a multiset \(\Gamma\) of roots in \(\Phi^+,\) let \(\kostantP_\Gamma(\nu)\) be the corresponding Kostant partition function, defined by \[\prod_{\gamma \in \Gamma} \frac{1}{1-\xvec^{-\gamma}} = \sum_{\nu} \kostantP_\Gamma(\nu)\xvec^{-\nu}.\] This is a vector partition function in the sense of [Stu95].

Fix a word \(i_1,\dotsc,i_\ell\) such that \[\key_\alpha(\xvec)=\pi_{i_\ell}\dotsm\pi_{i_1}(\xvec^\lambda),\] where \(\lambda=\sort(\alpha).\) Writing \(\delta_i=\evec_i-\evec_{i+1},\) the operator identity \[\pi_i(F)= \frac{F}{1-\xvec^{-\delta_i}} - \frac{\xvec^{-\delta_i}s_i(F)}{1-\xvec^{-\delta_i}}\] lets us expand one Demazure operator at a time. If a reflected denominator root becomes negative, rewrite it using \[\frac{1}{1-\xvec^\gamma} = -\frac{\xvec^{-\gamma}}{1-\xvec^{-\gamma}}.\] After all roots are made positive, this gives a finite signed list of triples \((c,\beta,\Gamma)\) with \[\key_\alpha(\xvec) = \sum_{(c,\beta,\Gamma)} c\,\xvec^\beta \prod_{\gamma \in \Gamma}\frac{1}{1-\xvec^{-\gamma}}.\]

#Representation theory of \(GL_n\)

For representation-theoretic background, see the introduction of [FGPS20].

V. Lakshmibai, P. Littelmann, and P. Magyar prove that the crystal of standard tableaux for a generalized Demazure module is a disjoint union of ordinary Demazure crystals [LLM02, Thm. 2]. Their Corollary 10 gives the corresponding nested isobaric-Demazure operator formula for its character. More explicitly, their tensor product rule decomposes the tensor product of a highest-weight line with a Demazure module into Demazure components indexed by the LS paths whose concatenations remain dominant [LLM02, Cor. 10, Prop. 12].

#Lascoux’s tableauhedron

A. Lascoux uses the name tableauhedron for a tableau/crystal construction that behaves like a weighted Eulerian incidence structure [Las12, pp. 13–24]. Let \(\lambda\) be a strict partition and choose a reduced word \(w=s_{i_1}\dotsm s_{i_\ell}.\) On the crystal of tableaux of shape \(\lambda,\) the reflection \(s_i\) reverses an \(i\)-string, while \(\pi_i\) sends the head of that string to the sum of all its elements. Lascoux combines these actions in \[d_i=\overline{\pi}_i\otimes s_i+1\otimes\pi_i =\pi_i\otimes s_i+1\otimes\theta_i, \qquad \theta_i=\overline{\pi}_i-s_i=\pi_i-1-s_i.\] The expansion of \(d_{i_1}\dotsm d_{i_\ell}\) records tableaux \(T\) with prescribed endpoints: their left key \(K_-(T)\) and right key \(K_+(T).\) In particular, the terms ending at \(w\) have \(K_+(T)=\lambda w,\) while the first tensor factor records \(K_-(T).\)

For key labels \(\sigma\) and \(\zeta\) in the orbit of \(\lambda,\) define the weighted incidence matrix \[\mathcal E[\sigma,\zeta] \coloneqq \sum_{\substack{T:\ K_-(T)=\sigma\\K_+(T)=\zeta}} \xvec^{\operatorname{wt}(T)}.\] Filter \(\mathcal E=\mathcal E_0+\mathcal E_1+\dotsb\) by the distance between the two keys. If \(f^\clubsuit(x_1,\dotsc,x_n)=f(x_1^{-1},\dotsc,x_n^{-1}),\) then Lascoux’s weighted Euler relation is [Las12, p. 22] \[\mathcal E^{-1} =\mathcal E_0^\clubsuit-\mathcal E_1^\clubsuit +\mathcal E_2^\clubsuit-\mathcal E_3^\clubsuit+\dotsb.\]

A compatible chain of tableaux is a sequence \(T_1,\dotsc,T_m\) satisfying \[K_+(T_j)\leq K_-(T_{j+1}) \qquad(1\leq j\lt{}m).\] These chains are the standard-monomial objects behind the Hilbert function \[L_{\lambda,w}(m)\coloneqq \key_{m\lambda,w}(1^n).\] Writing \(d=\deg_m L_{\lambda,w}(m)\) defines the key–Ehrhart numerator \(E_{\lambda,w}(z)\) by \[\sum_{m\geq0}L_{\lambda,w}(m)z^m =\frac{E_{\lambda,w}(z)}{(1-z)^{d+1}}.\] Lascoux proposes the tableauhedron as a structure for understanding this positive Hilbert numerator, while noting that its direct description in tableauhedron terms remains to be written [Las12, pp. 24–26]. The same dilation count has a separate union-of-GT-faces model.

Despite its name, the tableauhedron in this source is not presented as an established convex polytope. There is also an essential reduced-word caveat: the operators \(d_i\) do not satisfy the braid relations. Consequently, the raw graph or incidence presentation may depend on the chosen reduced word. The Euler relations explain cancellations after evaluation, but the source does not construct a reduced-word-independent Euler quotient, much less a canonical polytope [Las12, pp. 14, 19–22].

#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].

G. Z. D. e. Moura’s senior thesis surveys operator, tableau, plactic, and crystal models for key and atom polynomials [Mou24]. The product-coefficient problem is a central motivation.

#The Reiner–Shimozono conjecture

The Reiner–Shimozono conjecture predicted that every product of two key polynomials expands positively into Demazure atoms. R. Hodges constructs an infinite family of Reiner–Shimozono counterexamples . For compositions \(a,b,c\) in \(4(p+q-1)\) variables, his family satisfies \[[\atom_c](\key_a\key_b) = N(p+q-1,p)\bigl(2-(p-2)(q-2)\bigr),\] where \(N(r,k)\) is a Narayana number [Hod26, Thm. 1.1]. Thus the coefficient is negative exactly when \((p-2)(q-2)\gt{}2\); the first examples in the family occur in 28 variables and have coefficients \(-105\) and \(-350.\) The three-variable positivity theorem of A. Pun remains valid [Pun16].

The same counterexamples disprove Polo’s conjecture that tensor products of section modules on Schubert varieties admit Schubert filtrations [Hod26, Cor. 1.2]. Hodges also gives an exact coefficient formula and a polynomial-time sufficient test for positivity on a large family of atom coefficients. The \(K\)-theoretic consequence is recorded under products and Lascoux atoms.

T. C. Miller proves a positive combinatorial rule for multiplying a permuted-basement Demazure atom by a Schur polynomial [Mil25]. The rule is given by a compatible pair of vertex models for the atom and the Schur factor, and it specializes to product rules for ordinary Demazure atoms and characters. R. Paten and D. Woodruff extend several Schur-positivity results to key positivity by replacing Schur polynomials with flagged Schur polynomials [PW26]. In particular, they prove key positivity for many Temperley–Lieb immanants of flagged Jacobi–Trudi matrices, give a key-expansion rule for most products of flagged skew Schur polynomials, and prove a related log-concavity result using Demazure crystals.

#Pieri rule for key polynomials

A Pieri rule for key polynomials, described via an Robinson–Schensted–Knuth-type algorithm, is given by S. Assaf and D. Quijada in [AQ19]. Their rule expands the product of an arbitrary key polynomial with a single-part key polynomial in the key basis. Unlike general key products, this Pieri case admits a cancellation-free, multiplicity-free formula. The proof replaces ordinary row insertion with an insertion algorithm on Kohnert diagrams.

N. Cape and S. Zemel study generating series of key polynomials in a modified Demazure algebra [CZ24]. For a fixed permutation \(w,\) the generating function \(K_w(\xvec,\tvec)=\sum_\lambda K_{\lambda,w}(\xvec)\tvec^\lambda\) is a rational function whose denominator is described by bounded ascending sequences determined by the first values of \(w\) [CZ24, Thm. 3.9]. The numerator has structured quadratic terms governed by multisets with multiple presentations as sums of such bounded sequences.

#Skew key polynomials

In [AW19], 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 V. Reiner and M. Shimozono [RS95]. Combining this key-positive expansion with the positive expansion of keys into Demazure atoms gives an atom-positive expansion of skew key polynomials.

#Complete flagged homogeneous polynomials

H. Ehrhard [Ehr23] introduces complete flagged homogeneous polynomials . For a weak composition \(a=(a_1,a_2,\dotsc),\) they are defined by \[h_a(\xvec) \coloneqq \prod_{i\geq 1} \completeH_{a_i}(x_1,\dotsc,x_i).\] They form a \(\setZ\)-basis for the polynomial ring, and their stable limit is the usual complete homogeneous symmetric function \(\completeH_{\sort(a)}.\)

Ehrhard gives a nonnegative expansion of these polynomials in key polynomials using a flagged version of the RSK correspondence. Since keys are atom-positive, the complete flagged homogeneous polynomials are also Demazure atom-positive. Conversely, key polynomials have a signed expansion in the complete flagged homogeneous basis.

#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 A. Lascoux and M.-P. Schützenberger in [LS90]. Mason’s semistandard augmented filling model gives a positive monomial expansion [Mas09]. They also expand positively in the fundamental particle basis [Sea19, HN24]. The sourced skew analogue recorded here is the skew key polynomial, rather than a separate skew Demazure atom family. B. Brubaker, V. Buciumas, D. Bump, and H. P. A. Gustafsson construct solvable colored five-vertex lattice models whose partition functions are type \(A\) Demazure atoms [BBBG21]. Their proof uses a colored Yang–Baxter equation, and the construction also gives algorithms for computing Lascoux–Schützenberger keys. 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 \(|i-j| \geq 2.\) However, \(\theta_i^2 = 0,\) so we must now pay extra attention to choices of permutations.

Let \(\lambda\) be a partition with at most \(n\) parts, and let \(\sigma \in \symS_n\) be a minimal coset representative in \(\symS_n / Stab_n(\lambda).\) The Demazure atom \(\atom_{\lambda,\sigma}(z)\) is defined as \[\atom_{\lambda,\sigma}(z) \coloneqq \theta_\sigma\left( z_1^{\lambda_1} \dotsm z_n^{\lambda_n} \right).\] Again, instead of using \((\lambda,\sigma)\) as an index, we use \(\sigma(\lambda)\) as an index. Here, we must choose \(\sigma\) to be the shortest permutation \(\sigma\) that sends \(\lambda\) to \(\alpha = \sigma(\lambda).\)

For example, if \(\alpha = 1113,\) then \(\lambda=3111,\) and we must pick \(\sigma = 4123,\) and \[\atom_{1113}(z) = \atom_{3111,4123}(z) = z_1 z_2 z_3 z_4^3 + z_1 z_2 z_3^2 z_4^2 + z_1 z_2^2 z_3z_4^2 + z_1^2 z_2 z_3 z_4^2.\] Similarly, \[\atom_{103}(z) = \atom_{310,312}(z) = z_1^3 z_2 + z_1^3 z_3 + z_1^2 z_2^2 + z_1^2 z_3^2 + z_1^2 z_2 z_3 + z_1 z_2^3 + z_1 z_3^3 + z_1 z_2 z_3^2 + z_1 z_2^2 z_3.\]

Since \(\pi_i = \theta_i+1,\) it follows that key polynomials are positive in the Demazure atom basis. Indeed, [Pun16, Thm. 2.8] gives \[\key_{\alpha}(z) = \sum_{\beta \leq \alpha} \atom_{\beta}(z),\] where we have \(\beta \leq \alpha\) if one can obtain \(\beta\) from \(\rev(\alpha)\) by successively permuting entries which are in increasing order. For example, \[\key_{301} = \atom_{103} + \atom_{130}+\atom_{301}+\atom_{310}\] since \(103,\) \(130,\) \(301\) and \(310\) can be obtained from \(\rev(301)=103\) by successive swaps of elements (see also Bruhat order). Similarly, \[\key_{002} = \atom_{200} + \atom_{020} + \atom_{002}.\] The notation used here is not uniform in the literature.

See A. Pun’s PhD thesis for a good overview of Demazure atoms and key polynomials [Pun16].

S. Armon studies atom-positivity through Demazure crystals [Arm23]. The paper gives a recursive formula for crystal Demazure atoms, generalizing a result of Lascoux–Schützenberger, and separates the notions of extremal subsets and atom-positive subsets in the crystal setting.

#Cauchy kernel

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

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].

E. Feigin, A. Khoroshkin, and I. Makedonskyi [FKM24] extend this circle of Cauchy identities from triangular matrices to arbitrary staircase-shaped matrices. Their standard filtrations give positive Cauchy identities whose terms are products of key polynomials and Demazure atoms, as well as an alternating-sign identity involving products of key polynomials.

L. Pena uses ordinary RSK to treat matrices supported on an arbitrary skew Ferrers shape [Pen26]. The outer boundary gives an upper bound on the right key of the insertion tableau, while the inner boundary gives a lower bound on its left key; both bounds depend on the recording tableau. This yields positive Cauchy expansions in Demazure atoms and Lascoux–Schützenberger standard basis polynomials, indexed by Bruhat intervals [Pen26, Thm. 25 and Cor. 50]. A specialization recovers the Cauchy identity for \(m\)-symmetric Schur functions at \(t=0.\)

E. Feigin, I. Makedonskyi, and D. Orr interpret the \(t=0\) specialization of the Mimachi–Noumi nonsymmetric \(q\)-Cauchy identity representation-theoretically [FMO23]. They realize the product side as a character of functions on a matrix space and identify the associated graded pieces with tensor products of generalized global Weyl modules for the Iwahori algebra.

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

See also [AGL22] for the connection between the Cauchy kernel, RSK, crystals and last passage percolation.

#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 \(D.\)

Let \(s_i\) interchange \(x_i\) and \(x_{i+1}\) for \(1 \leq i \lt{} n,\) let \(s_n\) send \(x_n\) to \(x_n^{-1},\) and let \(\tau_n\) send \((x_{n-1},x_n)\) to \((x_n^{-1},x_{n-1}^{-1}).\) Fu and Lascoux write operators on the right. With \(f^w\) denoting the action of \(w\) on \(f,\) their isobaric divided differences are \[\begin{aligned} f\pi_i &= \frac{x_i f-x_{i+1}f^{s_i}}{x_i-x_{i+1}}, &&1 \leq i \lt{} n,\\ f\pi_n^C &= \frac{x_n f-x_n^{-1}f^{s_n}}{x_n-x_n^{-1}},\\ f\pi_n^B &= \frac{x_n f-f^{s_n}}{x_n-1},\\ f\pi_n^D &= \frac{f-x_{n-1}^{-1}x_n^{-1}f^{\tau_n}} {1-x_{n-1}^{-1}x_n^{-1}}. \end{aligned}\] They also use the second family \(\widehat{\pi}_i\coloneqq\pi_i-1\) and \(\widehat{\pi}_n^\heartsuit\coloneqq\pi_n^\heartsuit-1,\) for \(\heartsuit=B,C,D.\) The \(\pi\)-operators give the type \(B,\) \(C,\) and \(D\) Demazure characters by applying products of operators to dominant monomials, while the hatted operators give the corresponding atom-like family [FL09, Sec. 3].

In particular, the type \(C\) key polynomials contain the symplectic Schur functions. There are operator, crystal-graph, and key-tableau definitions of these polynomials.

See also [RY11], since key polynomials are specializations of nonsymmetric Macdonald polynomials.

#Type \(B/C\) key polynomials

In [San21], J. M. Santos gives combinatorial formulas for the type \(C\) key polynomials and Demazure atom polynomials. The type \(C\) key polynomials expand positively into type \(C\) Demazure atoms, by summing over an ideal in the Bruhat poset; see [San21, Prop. 53]. This is analogous to the type \(A\) setting. A crystal characterization is also given, as well as analogues of key tableaux.

V. Buciumas and T. Scrimshaw construct quasi-solvable lattice models for the \(\mathrm{Sp}_{2n}\) and \(\mathrm{SO}_{2n+1}\) Demazure atoms and characters [BS22]. This extends the vertex-model viewpoint on keys and atoms beyond type \(A.\)

#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.

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