The symmetric functions catalog

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2023-09-20

Grothendieck polynomials

The Grothendieck polynomials are closely related to the Schubert polynomials. They were introduced by A. Lascoux, 1990 in [Las90].

In [BFHTW20], the authors introduce biaxial Grothendieck polynomials, which generalize the double Grothendieck polynomials as well as the double dual Grothendieck polynomials. They prove several results regarding Grothendieck polynomials, such as branching rules and Cauchy identities. They also provide a solvable lattice model whose partition function is given by the Grothendieck polynomials, and duals.

Operator definition

Define $\partial_i(f)$ as $\frac{f-s_i(f)}{x_i - x_{i+1}}$ for $i=1,\dotsc,n-1,$ and $\pi_i \coloneqq \partial_i(1-x_{i+1}).$

The Grothendieck polynomials are then defined as

\begin{equation*} \grothendieck_\omega(x_1,\dotsc,x_n) \coloneqq \pi_{\omega^{-1}\omega_0}(x_1^{n-1}x_2^{n-2}\dotsm x_{n-1}). \end{equation*}

Note that the lowest degree homogenous part of $\grothendieck_\omega(\xvec)$ is given by the Schubert polynomial $\schubert_\omega(\xvec).$

There is a connection with polytopes just as for Schubert polynomials, see [MD17].

Pipedreams

As with Schubert polynomials, there is a way to describe Grothendieck polynomials using pipe dreams, see [FK94KM04]. The double Grothendieck polynomials are given as

\[ \grothendieck^{(\beta)}_\omega(\xvec,\yvec) = \sum_{P \in PD(\omega)} \beta^{-\length(\omega)} \prod_{(i,j)\in S(P)} \beta x_i \oplus y_j \]

where we sum over pipe dreams and $a \oplus b = a+b+\beta ab.$ Here, $S(P)$ is the set of crosses the pipe dream (this particular version of the formula is from [Thm.6.1, Wei21]).

A more recent formula expresses the the double Grothendieck polynomials as a sum over bumpless pipe dreams [Thm.1.1, Wei21]:

\[ \grothendieck^{(\beta)}_\omega(\xvec,\yvec) = \beta^{-\length(\omega)} \sum_{P \in BPD(\omega)} \prod_{(i,j)\in D(P)} \beta (x_i \oplus y_j ) \prod_{(i,j)\in NW(P)} 1+ \beta (x_i \oplus y_j ) \]

where $BPD(\omega)$ is the set of bumpless pipe dreams of type $\omega$ and $NW(P)$ is the set of West-to-North pipe bends.

However, this formula does not immediately give the same pipe dream expression, as the Fomin–Kirillov formula, and it is an open problem to find a bijection between the two models.

A lattice model is also given in [BS20], and a connection with bumpless pipe dreams is given.

Alternating sign matrices

There is an expression for Grothendieck polynomials using alternating sign matrices, due to A. Lascoux.

Stable Grothendieck polynomials

The stable Grothendieck polynomials were introduced by S. Fomin and A. Kirillov in [FK94FK96b]. Similar to Stanley symmetric functions, the stable Grothendieck polynomials are defined as the limit

\begin{equation*} \grothendieckStable_\omega(\xvec) \coloneqq \lim_{k\to \infty} \grothendieck_{1^k \times \omega}(\xvec). \end{equation*}

Grassman Grothendieck polynomials

See this page for the definition of a Grassman permutation, and how to associate a partition with such a permutation. For a nice recent paper with lots of references and results, see [MPS18].

Stable Grothendieck polynomials indexed by Grassman permutations are symmetric polynomials. They serve as a K-theoretic analog of the Stanley symmetric functions. These can be computed using a combinatorial formula, using set-valued tableaux:

\begin{align} \grothendieckStable_\lambda(\xvec) = \sum_{T \in \mathrm{SVT}(\lambda)} (-1)^{|T|-|\lambda|} \xvec^T. \end{align}

The set $\mathrm{SVT}(\lambda)$ consists of all set-valued tableaux of shape lambda, where each box is filled with a non-empty set of natural numbers. Rows must be weakly increasing and column must be strictly increasing. The sum of the sizes of the sets in $T$ is denoted $|T|.$

A Gelfand–Tsetlin pattern type combinatorial formula is described in [MS14], where marked GT-pattern are used to give the monomial expansion of $\grothendieckStable_\lambda.$ See also [MPS18], where they give an alternative proof of this.

Weyl identity

A Weyl-type bialternant formula is given by the following quotient, see [IN13Kir16Iwa19].

\begin{align} \grothendieckStable_\lambda(x_1,\dotsc,x_n) = \frac{ \det \left( x_i^{\lambda_j+n-j}(1+\beta x_i)^{j-1} \right)_{1\leq i,j \leq n} }{ \prod_{1 \leq i \lt j \leq n} (x_i - x_j) } \qquad (\beta = -1). \end{align}

Jacobi–Trudi identity

There is a Jacobi–Trudi type identity, [Thm. 1.10, Kir16] given by

\begin{align} \grothendieckStable_\lambda(\xvec) = \det\left( \sum_{m=0}^\infty \binom{i-1}{m} \beta^m \completeH_{\lambda_i-i+j+m}(\xvec) \right)_{1\leq i,j \leq n}. \end{align}

Pieri rule

C. Lenart [Len00] proved the following Pieri rule for stable Grassman Grothendieck polynomials:

\[ \grothendieckStable_{(a)}\grothendieckStable_{\lambda} = \sum_{\mu / \lambda \text{ horizontal strip}} (-1)^{|\mu/\lambda|-a} \binom{ \textrm{rows}(\mu/\lambda)-1 }{ |\mu/\lambda|-a } \grothendieckStable_{\mu} \]

where $\textrm{rows}(\mu/\lambda)$ is the number of rows in the horizontal strip.

Murnaghan–Nakayama rule

In [TDHH21], the following Murnaghan–Nakayama type rule is the main result.

\begin{align} \grothendieckStable_\lambda(\xvec) \powerSum_k(\xvec) = \sum_{\mu} (\beta)^{|\mu|-|\lambda|-k} (-1)^{k-c(\mu/\lambda)} \binom{r(\mu/\lambda)-1}{k-c(\mu/\lambda)} \grothendieckStable_\mu(\xvec) \end{align}

where the sum runs over all connected skew shapes $\mu/\lambda,$ containing at least one $k$-ribbon along its northwestern border. Here, $c(\mu/\lambda)$ is the number of columns spanned by the shape. Likewise, $r(\mu/\lambda)$ is the number of rows spanned by the shape.

Note that at $\beta=0,$ we recover the classical Murnaghan–Nakayama rule for Schur polynomials.

Schur expansion

C. Lenart [Thm. 2.8, Len00] prove that whenever $k\geq \length(\lambda),$

\[ \grothendieckStable_{\lambda}(x_1,\dotsc,x_k) = \sum_{\lambda \subseteq \mu \subseteq \hat{\lambda}} (-1)^{|\mu|-|\lambda|} a_{\lambda\mu} \schurS_{\mu}(\xvec) \]

where the sum is taken over partitions $\mu,$ \hat{\lambda} is the unique maximal partition of length $k$ obtained from $\lambda$ by adding at most $j-1$ boxes to the $j$-th row of the Young diagram $\lambda.$

The coefficients $a_{\lambda\mu}$ are non-negative, and count the number of increasing tableaux of shape $\mu/\lambda.$ These are tableaux which are strictly increasing along rows and columns, and for each $i\geq 1,$ row $i$ only contain entries $\leq i-1.$

Alternatively, in [Cor. 3.11, MPS18], it is proved that $a_{\lambda\mu}$ count the number of set-valued tableaux of shape $\lambda$ and weight $\mu,$ which are also Yamanouchi. The authors archieve this by defining a type $A$ crystal graph on set valued fillings.

Littlewood–Richardson rule

There is a Littlewood–Richardson rule for the product of stable Grothendieck polynomials:

\[ \grothendieckStable_{\lambda}(\xvec;\yvec) \grothendieckStable_{\mu}(\xvec;\yvec) = \sum_{\nu} (-1)^{|\lambda|+|\mu|-|\nu|} c_{\lambda\mu}^{\nu} \grothendieckStable_{\nu}(\xvec;\yvec) \]

where $c_{\lambda\mu}^{\nu}$ is the number of set-valued tableaux of shape $\lambda * \mu$ with content $\nu.$

Skew stable Grothendieck

In [Buc02LMS16], the skew version of the stable Grothendieck polynomials are defined as

\begin{align} \grothendieckStable_{\nu/\lambda}(\xvec) = \sum_{\mu} \sum_{T \in \mathrm{SVT}(\nu/\lambda,\mu)} (-1)^{|\lambda|-|\nu|-|\mu|} \monomial_\mu(\xvec). \end{align}

From a result in [BJS93], it follows that the skew stable Grothendieck polynomials are precisely the stable limit of Grothendieck polynomials indexed by 321-avoiding permutations.

A recent result in [CP19] gives an expansion of the skew stable Grothendieck polynomials in terms of skew Schur polynomials. Here, $\sigma$ and $\mu$ are skew shapes.

\begin{align} \grothendieckStable_{\sigma}(\xvec) = \sum_{\mu \supseteq \sigma} (-1)^{|B(\mu/\sigma)|} a_{\sigma,\mu} \schurS_\mu(\xvec). \end{align}

The motivation behind this formula is the connection with Euler characteristics of certain varieties, see [CP17].

The coefficients $a_{\sigma,\mu}$ are non-negative integers with an explicit combinatorial interpretation. M. Chan and N. Pflueger also prove a similar inverse formula, for connected skew shapes $\sigma$:

\begin{align} \schurS_{\sigma}(\xvec) = \sum_{\mu \supseteq \sigma} (-1)^{|A(\mu/\sigma)|} b_{\sigma,\mu} \grothendieckStable_\mu(\xvec). \end{align}

More information

See this paper for the expansion of stable Grothendieck polynomials into Grassman Grothendieck polynomials. In https://arxiv.org/pdf/1701.03561.pdf, a flagged version is considered. See https://arxiv.org/pdf/1711.09544.pdf for several Cauchy identities, and skew versions.

Problem (See [MPS18]).

Describe a crystal structure, whose connected components give stable Grothendieck polynomials. This would be analogous to the type $A$ crystals that produce Schur polynomials.

Double Grothendieck polynomials

The double Grothendieck polynomials are defined via divided difference operators, $\pi_i,$ defined as

\[ \pi_i \coloneqq \frac{ (1+\beta x_{i+1})f-(1+\beta x_i)s_i(f)}{x_i - x_{i+1}}, \]

where $s_i$ permutes $x_i$ and $x_{i+1}.$ We then have the base case

\[ \grothendieck_{\omega_0}(x|b) = \prod_{i+j \leq n}(x_i + b_j + \beta x_i b_j), \]

and whenever $\length(\omega s_i) = 1 + \length(\omega)$ the recursive definition

\[ \grothendieck_{\omega}(x|b) \coloneqq \pi_i \left( \grothendieck_{\omega s_i}(x|b) \right). \]

Canonical stable Grothendieck polynomials

In [Yel17], the author introduce a two-parameter deformation of the stable Grothendieck polynomials, $\grothendieckStable^{(a,b)}_\lambda(\xvec)$ and the dual stable Grothendieck polynomials, $\grothendieckDual^{(a,b)}_\lambda(\xvec)$ with the following properties.

  1. Specializes to Schur polynomials, $\schurS_\lambda(\xvec) = \grothendieckStable^{(0,0)}_\lambda(\xvec),$
  2. Specializes to classical Grothendieck polynomials, $\grothendieckStable(\xvec) = \grothendieckStable^{(0,-1)}_\lambda(\xvec)$ and $\grothendieckDual(\xvec) = \grothendieckDual^{(0,1)}_\lambda(\xvec),$
  3. Fulfill $\omega(\grothendieckStable(\xvec)) = \grothendieckStable^{(-1,0)}_{\lambda'}(\xvec)$ and $\omega(\grothendieckDual(\xvec)) = \grothendieckDual^{(1,0)}_{\lambda'}(\xvec),$
  4. Highest (lowest) degree term is a Schur polynomial, \[ \grothendieckStable^{(a,b)}_\lambda(\xvec) = \schurS_\lambda(\xvec) + \text{higher terms} \quad \grothendieckDual^{(a,b)}_\lambda(\xvec) = \schurS_\lambda(\xvec) + \text{lower terms} \]
  5. They are dual with respect to the Hall inner product, $ \langle \grothendieckStable^{(-a,-b)}_\lambda, \grothendieckDual^{(a,b)}_\mu \rangle = \delta_{\lambda\mu} $
  6. Nice involution properties, \[ \omega( \grothendieckStable^{(a,b)}_\lambda ) = \grothendieckStable^{(b,a)}_{\lambda'} \text{ and } \omega( \grothendieckDual^{(a,b)}_\lambda ) = \grothendieckDual^{(b,a)}_{\lambda'} \]
  7. They have the same structure constants $c^{\nu}_{\lambda\mu}$ as the usual stable Grothendieck polynomials, \[ \grothendieckStable^{(a,b)}_\lambda \grothendieckStable^{(a,b)}_\mu = \sum_{\nu} (a+b)^{|\nu|-|\lambda|-|\mu|} c^{\nu}_{\lambda\mu}\grothendieckStable^{(a,b)}_\nu. \]

There is a skew version of these, a Weyl-type alternant formula, and many other nice properties, such as tableau descriptions, operator descriptions etc.

The $\grothendieckDual^{(a,b)}_\mu $ are Schur-positive, and a crystal structure on the canonical Stable Grothendieck polynomials, as well as their duals is given by G. Hawkes and T. Scrimshaw [HS19].

In [GZ20], solvable vertex models are used to prove Cauchy identities for the canonical Grothendieck polynomials:

\[ \sum_{\lambda} \grothendieckStable^{(-a,-b)}_\lambda(x_1,\dotsc,x_m) \grothendieckDual^{(a,b)}_\lambda(y_1,\dotsc,y_n) = \prod_{\substack{ 1 \leq i \leq m \\ 1 \leq j \leq n }} \frac{1}{1 - x_i y_j} \] \[ \sum_{\lambda} \grothendieckStable^{(-a,-b)}_{\lambda'}(x_1,\dotsc,x_m) \grothendieckDual^{(a,b)}_\lambda(y_1,\dotsc,y_n) = \prod_{\substack{ 1 \leq i \leq m \\ 1 \leq j \leq n }} (1 + x_i y_j ). \]

Quantum double Grothendieck polynomials

In [LM06], the authors introduce the quantum double Grothendieck polynomials.

Flagged factorial Grothendieck polynomials

In [MS19], the authors consider a generalization of the Grassman Grothendieck polynomials. They introduce the flagged factorial Grothendieck polynomials, indexed by a partition $\lambda$ with $r$ parts and a flag $f = (0 \lt f_1 \leq f_2 \leq \dotsb \leq f_r).$ We then let

\[ \grothendieckStable_{\lambda,f}(\xvec|b) \coloneqq \sum_{T \in SVT(\lambda,f)} \beta^{|T|-|\lambda|}[x|b]^T \]

where the sum runs over all flagged set-valued tableaux of shape $\lambda$ and flag $f.$ This means that the sets in row $i$ may only contain elements from $\{f_1,\dotsc,f_i\}.$ Furthermore,

\[ [x|b]^T \coloneqq \prod_{e\in T} x_{val(e)} \oplus b_{val(e)-r(e)+c(e)}. \]

Here, $a\oplus b \coloneqq a+b+\beta ab.$

In[MS19], the authors give a Jacobi–Trudi type identity for flagged factorial Grothendieck polynomials, and show that these are equal to double Grothendieck polynomials for vexillary ($2143$-avoiding) permutations.

Dual stable Grothendieck polynomials

The dual stable Grothendieck polynomials were first studied in [LP07]. They are defined through the Hall inner product

\[ \langle \grothendieckDual_\lambda, \grothendieckStable_\mu \rangle = \delta_{\lambda \mu}. \]

Let $\mathrm{RPP}(\lambda)$ be the set of fillings of $\lambda$ with weakly increasing rows and columns. Then

\[ \grothendieckDual_\lambda(\xvec) \coloneqq \sum_{T \in \mathrm{RPP}(\lambda)} \xvec^{ev(T)} \]

where $ev(T)_i$is the number of columns of $T$ where $i$ appears. Note that

\[ \grothendieckDual_\lambda(\xvec) = \schurS_\lambda(\xvec) + \text{lower order terms}. \]

The skew version $\grothendieckDual_{\lambda/\mu}(\xvec)$ is defined in a similar manner. One can prove that $\grothendieckDual_{\lambda/\mu}(\xvec)$ is symmetric by using a version of the Bender–Knuth involutions, see [GGL16].

In [LMS16], it is shown that the constants appearing in the expansion

\[ \grothendieckDual_{\nu/\mu}(\xvec) = \sum_{\nu} (-1)^{|\lambda|+|\mu|-|\nu|} c^{\nu}_{\mu\lambda} \grothendieckDual_{\lambda}(\xvec), \]

agree with the coefficients $c_{\lambda\mu}^{\nu}$ in the Littlewood–Richardson rule for stable Grothendieck polynomials.

Bialternant formula

A bialternant formula for the dual stable Grothendieck polynomials is proved in [Thm. 17, Yel19]. It states that

\begin{align} \grothendieckDual_{\lambda}(\xvec) = \frac{1}{ \prod_{i \lt j} (x_i-x_j)} \det\left[ \phi^{i-1}x_j^{\lambda_i+n-i} \right]_{1\leq i,j \leq n} \end{align}

where $\phi^k x^n = \sum_{j=0}^n \binom{k+j-1}{j}x^{n-j}.$

Littlewood-type identities

Analogous to the Schur function Littlewood identities, we have

\[ \sum_{\lambda : \length(\lambda)\leq n} \grothendieckDual_\lambda(x_1,\dotsc,x_m) = \prod_{i=1}^m \frac{1}{(1-z_i)^n}. \]

Jacobi–Trudi identity

There is a Jacobi–Trudi type identity, [Prop. 4.4, Iwa19] given by

\begin{align} \grothendieckDual_\lambda(\xvec) = \det\left[ \sum_{m=0}^\infty \binom{1-i}{m} \beta^m \completeH_{\lambda_i-i+j-m}(\xvec) \right]_{1\leq i,j \leq n}. \end{align}

Alternatively, [Yel17AY20], we have

\begin{align} \grothendieckDual_\lambda(\xvec) = \det\left[ \completeH_{\lambda_i-i+j}(1^{\lambda_i-1},\xvec) \right]_{1\leq i,j \leq \length(\lambda)} = \det\left[ \elementaryE_{\lambda'_i-i+j}(1^{\lambda'_i-1},\xvec) \right]_{1\leq i,j \leq \lambda_1}. \end{align}

There is also a Jacobi–Trudi identity for the skew dual stable Grothendieck polynomials, originally conjectured by D. Grindberg (see these slides). A proof was later presented in Y.-S Kim [Kim20]. A different proof was obtained independently by A. Amanov and D. Yeliussizov [AY20]. Both approaches use quite novel techniques with non-intersecting lattice paths.

The skew Jacobi–Trudi identity states that

\begin{align} \grothendieckDual_{\lambda/\mu}(\xvec) = \det\left[ \elementaryE_{\lambda'_i-i-\mu'_j+j}(1^{\lambda'_i-\mu'_j-1},\xvec) \right]_{1\leq i,j \leq \lambda_1}. \end{align}

The skew Jacobi–Trudi identity extends to the family of refined dual stable Grothendieck polynomials, introduced in [GGL16], see [Kim20AY20] for proofs.

Schur expansion

In [Thm. 9.8, LP07], the authors show that

\[ \grothendieckDual_\lambda(\xvec) = \sum_\mu f^\mu_\lambda \schurS_\mu(\xvec) \]

where $f^\mu_\lambda $ is the number of elegant fillings of shape $\lambda/\mu.$ A filling is elegant if it is semi-standard, and all entries in row $i$ is in $1,\dotsc,i-1.$ See also [Gal17] for a crystal proof.

A flagged version of the skew dual Grothendieck polynomials are shown to be key positive in [Kun23].

Last passage percolation

D. Yeliussizov [Yel20] show that the dual Grothendieck polynomials can be realized as a column distribution of a directed last-passage percolation model.

Generalized and refined dual stable Grothendieck polynomials

The refined dual stable Grothendieck polynomial were introduced in [GGL16]. They define these as

\[ \grothendieckRefinedDualStable_\lambda(\xvec;\tvec) \coloneqq \sum_{T \in RPP(\lambda)} \xvec^{ev(T)} \tvec^{b(T)} \]

where (as usual), $ev(T)_i$ is the number of columns containing $i,$ and $b(T)_j$ is the number of boxes in row $j$ whose entry is equal to the entry in box immediately below it.

In [Yel19], the generalized dual stable Grothendieck polynomials are defined:

\[ \grothendieckDual_\lambda(\xvec;\zvec) \coloneqq \sum_{\pi \in PP(\lambda)} \prod_{(i,j) \in \DES(\pi)} x_i z_{\pi(i,j)}. \]

Note that $\grothendieckDual_\lambda(1^n;\zvec) = \grothendieckDual_\lambda(\zvec).$ It is observed that $\grothendieckDual_\lambda(\xvec;\zvec) = \xvec^{\lambda}\grothendieckRefinedDualStable_\lambda(\xvec^{-1},\zvec),$ where $\grothendieckRefinedDualStable_\lambda$ is a refined dual stable Grothendieck polynomial.

D. Yeliussizov ([Eq. (11), Yel19]) prove that

\[ \sum_{\lambda : \length(\lambda)\leq n} \grothendieckDual_\lambda(x_1,\dotsc,x_m; z_1,\dotsc,z_n) = \prod_{i=1}^m \prod_{j=1}^n \frac{1}{1-x_iz_j}. \]

D. Yeliussizov proves several new $q$-identites involving plane partitions and dual Grothendieck polynomials in [Yel19].

In [MS20], the authors introduce a vertex model whose partition function is the refined dual stable Grothendieck polynomial. They give numerous new identities, and prove Jacobi–Trudi identities for the skew refined dual stable Grothendieck polynomials. Moreover, a new Cauchy identity is given as well. This article has a lot of interesting new identities and connections.

References