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Degrees of unitary Deligne–Lusztig varieties

Chao Li Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA chaoli@math.columbia.edu
Abstract.

We prove an explicit degree formula for certain unitary Deligne–Lusztig varieties. Combining with an alternative degree formula in terms of Schubert calculus, we deduce several algebraic combinatorial identities which may be of independent interest.

1. Introduction

1.1. Degrees of unitary Deligne–Lusztig varieties

Let k0=𝔽qk_{0}=\mathbb{F}_{q} be a finite field of size qq. Let k=𝔽q2k=\mathbb{F}_{q^{2}} be the quadratic extension of k0k_{0}. Let VV be a (nondegenerate) k/k0k/k_{0}-hermitian space of dimension nn. Let Grm(V)\operatorname{Gr}_{m}(V) be the Grassmannian of mm-dimensional subspaces of VV, which is a smooth projective variety over kk of dimension m(nm)m(n-m).

We assume that VV has dimension n=2d+1n=2d+1 (d0d\geq 0) over kk and take m=d+1m=d+1. Define DL(V)Grd+1(V)\operatorname{DL}(V)\subseteq\operatorname{Gr}_{d+1}(V) to be the closed kk-subscheme parametrizing (d+1)(d+1)-dimensional subspaces UU such that UUU^{\perp}\subseteq U, where UU^{\perp} is the orthogonal complement of UU in VV (see the more precise Definition 2.1). It is smooth, projective and geometrically irreducible of dimension dd, and is known as a unitary Deligne–Lusztig variety, as it can be identified as a generalized Deligne–Lusztig variety associated to a parabolic subgroup of the odd unitary group U(V)\mathrm{U}(V) ([VW11, §4.5]). This class of varieties shows up in the study of the supersingular locus of unitary Shimura varieties ([VW11]) and plays an important role in the arithmetic of unitary Shimura varieties, such as the Kudla–Rapoport conjecture ([KR11, KR14]).

Our first main result is a simple degree formula for DL(V)\operatorname{DL}(V). Recall that the degree of a projective variety XNX\subseteq\mathbb{P}^{N} of pure dimension dd is given by the geometric intersection number of XX with dd general hyperplanes in N\mathbb{P}^{N}. Denote by degDL(V)\deg\operatorname{DL}(V) the degree of DL(V)\operatorname{DL}(V) under the Plücker embedding Grd+1(V)(d+1V)\operatorname{Gr}_{d+1}(V)\hookrightarrow\mathbb{P}(\wedge^{d+1}V).

Theorem 1.1.

Let VV be a k/k0k/k_{0}-hermitian space of dimension n=2d+1n=2d+1. Then

(1.1.1) degDL(V)=i=1d1q2i1q.\deg\operatorname{DL}(V)=\prod_{i=1}^{d}{\frac{1-q^{2i}}{1-q}}.
Remark 1.2.

Here (and below) we regard the empty product (when d=0d=0) as 1. The right hand side of (1.1.1) can also be interpreted as the qq-analogue of the double factorial

[2d]q!!:=[2d]q[2d2]q[2]q,[2d]_{q}!!:=[2d]_{q}[2d-2]_{q}\cdots[2]_{q},

where [n]q=1qn1q[n]_{q}=\frac{1-q^{n}}{1-q} is the qq-analogue of nn.

Theorem 1.1 will be proved in §2.6. Its proof is inspired by the higher local modularity in the recent proof of the Kudla–Rapoport conjecture (cf. [LZ22a, §6.4]). In fact, the key formulas (Propositions 2.11, 2.12) can be extracted from certain vertical intersection formulas ([LZ22a, Lemmas 6.4.5, 6.4.6]) on unitary Rapoport–Zink spaces. To make the ideas more transparent, here we work directly on DL(V)\operatorname{DL}(V) and introduce the notions of special cycles Z(W)DL(V)Z(W)\subseteq\operatorname{DL}(V) (Definition 2.3) and a special line bundle DL(V)\mathcal{L}_{\operatorname{DL}(V)} (Definition 2.5) on DL(V)\operatorname{DL}(V). These notions may be viewed respectively as finite field analogues of special cycles and tautological line bundles on unitary Shimura varieties. From this perspective, the degree formula in Theorem 1.1, or more precisely the formula for c1(DL(V))dc_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})^{d} in Proposition 2.12, may be viewed as a finite field analogue of the constant term formula in Kudla’s geometric Siegel–Weil formula (relating the geometric volume of unitary Shimura varieties and an abelian LL-value, cf. [Kud04, (4.4)] for the analogue for orthogonal Shimura varieties).

The proof of Theorem 1.1 ultimately relies on identifying the special cycles Z(W)DL(V)Z(W)\subseteq\operatorname{DL}(V) as unitary Deligne–Lusztig subvarieties and the (proved) Tate conjecture for DL(V)\operatorname{DL}(V), in order to perform induction on the dimension dd. These inductive structures are available for Deligne–Lusztig varieties beyond those of unitary types (e.g., the type Dn2{}^{2}D_{n} considered in [LZ22b, §7.6] and two other types Bn,CnB_{n},C_{n} listed in [HLZ19, §3]), and it would be interesting to extend the method to obtain degree formulas for more general Deligne–Lusztig varieties.

1.2. Schubert calculus and applications to algebraic combinatorics

We also prove a different formula for degDL(V)\deg\operatorname{DL}(V) in terms of Schubert calculus on Grassmannians (see also the related general works [Kim20, HP20]).

Theorem 1.3.

The following identity holds in Chd(d+1)(Grd+1(V)k¯){\mathrm{Ch}}^{d(d+1)}(\operatorname{Gr}_{d+1}(V)_{\bar{k}})_{\mathbb{Q}}\simeq\mathbb{Q}:

(1.3.1) degDL(V)=cσcσc^σ1dq|c|,\deg\operatorname{DL}(V)=\sum_{c}\sigma_{c}\sigma_{\widehat{c}^{\prime}}\sigma_{1}^{d}q^{|c|},

where the sum runs over all integer tuples c=(c1,,cd)c=(c_{1},\ldots,c_{d}) with dc1cd0d\geq c_{1}\geq\cdots\geq c_{d}\geq 0. Here (as recalled in §3.1):

  • \bullet

    |c|=c1++cd|c|=c_{1}+\cdots+c_{d},

  • \bullet

    σcCh|c|(Grd+1(V)k¯)\sigma_{c}\in{\mathrm{Ch}}^{|c|}(\operatorname{Gr}_{d+1}(V)_{\bar{k}})_{\mathbb{Q}} is the Schubert class,

  • \bullet

    c^\widehat{c} is the complement of cc defined by c^:=(dcd,,dc1)\hat{c}:=(d-c_{d},\ldots,d-c_{1}),

  • \bullet

    c^\widehat{c}^{\prime} is the conjugate of c^\widehat{c}.

Combining Theorems 1.1 and 1.3 we deduce several combinatorial identities.

Corollary 1.4.
  • (i)

    The right hand side of (1.3.1) is equal to i=1d1q2i1q\prod_{i=1}^{d}{\frac{1-q^{2i}}{1-q}}.

  • (ii)

    The x12dx22d1xdd+1x_{1}^{2d}x_{2}^{2d-1}\cdots x_{d}^{d+1}-coefficient of

    (i,j=1d(qxi+xj))(x1++xd)d(i<j(xixj))\left(\prod_{i,j=1}^{d}(qx_{i}+x_{j})\right)(x_{1}+\cdots+x_{d})^{d}\left(\prod_{i<j}(x_{i}-x_{j})\right)

    is equal to i=1d1q2i1q\prod_{i=1}^{d}{\frac{1-q^{2i}}{1-q}}.

  • (iii)

    Let l0l\geq 0. The following two sets have the same size:

    • \bullet

      The set of standard Young tableaux of skew shape (c^)/c(\widehat{c}^{\prime})^{*}/c, where cc runs over (c1,,cd)(c_{1},\ldots,c_{d}) such that dc1cd0d\geq c_{1}\geq\cdots\geq c_{d}\geq 0 and |c|=l|c|=l. Here (c^)(\widehat{c}^{\prime})^{*} is the dual of c^\widehat{c}^{\prime} (recalled in §3.1).

    • \bullet

      The set of ordered partitions (l1,,ld)(l_{1},\ldots,l_{d}) of ll satisfying l=l1++ldl=l_{1}+\cdots+l_{d} and 0li2i10\leq l_{i}\leq 2i-1 for all 1id1\leq i\leq d.

Theorem 1.3 and Corollary 1.4 will be proved §3.3 and §3.4 respectively. We notice that the combinatorial identities in Corollary 1.4 seem to be quite nontrivial even for small values of dd (see Example 3.4). It would be very interesting to find more direct combinatorial proofs of these identities.

1.3. Acknowledgments

It is the author’s pleasure to dedicate this paper to Steve Kudla on the occasion of his 70th birthday. The influence of his original insights on the geometric and arithmetic Siegel–Weil formula on this paper should be evident to the readers. The author is also grateful to Z. Yun and W. Zhang for helpful conversations, and to M. Rapoport and the anonymous referee for useful comments. The author’s work is partially supported by the NSF grant DMS-2101157.

1.4. Notation

Let XX be a smooth projective variety over a finite field kk. Let Chr(Xk¯){\mathrm{Ch}}^{r}(X_{\bar{k}}) be the Chow group of codimension rr algebraic cycles of Xk¯X_{\bar{k}} defined modulo rational equivalence. Fix a prime char(k)\ell\neq\operatorname{char}(k), denote by

clr:Chr(Xk¯)H2r(Xk¯,)(r){\mathrm{cl}}_{r}:{\mathrm{Ch}}^{r}(X_{\bar{k}})_{\mathbb{Q}}\rightarrow\operatorname{H}^{2r}(X_{\bar{k}},\mathbb{Q}_{\ell})(r)

the \ell-adic cycle class map. Denote by Tate2r(Xk¯)H2r(Xk¯,)(r)\mathrm{Tate}^{2r}_{\ell}(X_{\bar{k}})\subseteq\operatorname{H}^{2r}(X_{\bar{k}},\mathbb{Q}_{\ell})(r) the subspace of Tate classes, i.e., the elements fixed by an open subgroup of Gal(k¯/k)\operatorname{Gal}({\bar{k}}/k). Then clr{\mathrm{cl}}_{r} intertwines the intersection product \cdot on the Chow ring and the cup product \cup on the cohomology ring, namely the following diagram commutes,

(1.4.1) Chr(Xk¯)clr×Chs(Xk¯)clsChr+s(Xk¯)clr+sH2r(Xk¯,)(r)×H2s(Xk¯,)(s)H2(r+s)(Xk¯,)(r+s).\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 32.6427pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\\&&&&\crcr}}}\ignorespaces{\hbox{\kern-23.33131pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{Ch}}^{r}(X_{\bar{k}})_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 0.0pt\raise-20.75333pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0pt\hbox{$\scriptstyle{{\mathrm{cl}}_{r}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.8689pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 35.64273pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\times}$}}}}}}}{\hbox{\kern 61.68102pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{Ch}}^{s}(X_{\bar{k}})_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 84.98381pt\raise-20.75333pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0pt\hbox{$\scriptstyle{{\mathrm{cl}}_{s}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 84.98381pt\raise-29.8689pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 125.61108pt\raise 4.55556pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.55556pt\hbox{$\scriptstyle{\cdot}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 150.87999pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 120.54709pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 150.87999pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{Ch}}^{r+s}(X_{\bar{k}})_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 177.7016pt\raise-20.75333pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.68056pt\hbox{$\scriptstyle{{\mathrm{cl}}_{r+s}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 177.7016pt\raise-29.57333pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-32.6427pt\raise-41.50665pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{H}^{2r}(X_{\bar{k}},\mathbb{Q}_{\ell})(r)}$}}}}}}}{\hbox{\kern 35.64273pt\raise-41.50665pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\times}$}}}}}}}{\hbox{\kern 52.42056pt\raise-41.50665pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{H}^{2s}(X_{\bar{k}},\mathbb{Q}_{\ell})(s)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 118.21376pt\raise-36.5622pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.94446pt\hbox{$\scriptstyle{\cup}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 129.54712pt\raise-41.50665pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 120.54709pt\raise-41.50665pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 129.54712pt\raise-41.50665pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{H}^{2(r+s)}(X_{\bar{k}},\mathbb{Q}_{\ell})(r+s).}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}

When r=1r=1, the cycle class map cl1:Ch1(Xk¯)H2(Xk¯,)(1){\mathrm{cl}}_{1}:{\mathrm{Ch}}^{1}(X_{\bar{k}})_{\mathbb{Q}}\rightarrow\operatorname{H}^{2}(X_{\bar{k}},\mathbb{Q}_{\ell})(1) is injective. When r=dimXr=\dim X, we often identify ChdimX(Xk¯){\mathrm{Ch}}^{\dim X}(X_{\bar{k}})_{\mathbb{Q}}\simeq\mathbb{Q} via the degree isomorphism (cf. [Ful98, Example 1.6.6]). Recall that the Tate conjecture ([Tat65, Conjecture 1], or [Tat94, Conjecture TiT^{i}]) asserts that for any r0r\geq 0, the image of \ell-adic cycle class map clr{\mathrm{cl}}_{r} \mathbb{Q}_{\ell}-spans Tate2r(Xk¯)\mathrm{Tate}^{2r}_{\ell}(X_{\bar{k}}).

For a subvariety ZXZ\subseteq X of pure codimension rr, we denote by [Z]X[Z]_{X} its class in Chr(Xk¯){\mathrm{Ch}}^{r}(X_{\bar{k}}). When the ambient variety XX is clear we suppress the subscript and simply write [Z][Z]. For any vector bundle 𝒱\mathcal{V} on XX, denote by cr(𝒱)Chr(Xk¯)c_{r}(\mathcal{V})\in{\mathrm{Ch}}^{r}(X_{\bar{k}}) its rr-th Chern class. By abusing notation we also denote by the same symbols [Z]X[Z]_{X}, [Z][Z] and cr(𝒱)c_{r}(\mathcal{V}) their images in Chr(Xk¯){\mathrm{Ch}}^{r}(X_{\bar{k}})_{\mathbb{Q}}. For any vector bundle 𝒱\mathcal{V} on XX, denote by 𝒱:=om𝒪X(𝒱,𝒪X)\mathcal{V}^{*}:=\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{V},\mathcal{O}_{X}) its dual bundle.

2. Degrees of unitary Deligne–Lusztig varieties

Let pp be a prime and let qq be a power of pp. Let k0=𝔽qk_{0}=\mathbb{F}_{q} be a finite field of size qq. Let k=𝔽q2k=\mathbb{F}_{q^{2}} be the quadratic extension of k0k_{0}. Denote by σ\sigma the absolute qq-Frobenius endomorphism on any scheme of characteristic pp.

2.1. Hermitian spaces

Let VV be a (nondegenerate) k/k0k/k_{0}-hermitian space of dimension nn, i.e., a kk-vector space of dimension nn equipped with a (nondegenerate) paring (,):V×Vk(\ ,\ ):V\times V\rightarrow k that is linear in the first variable, σ\sigma-linear in the second variable and satisfies

(2.0.1) (x,y)=(y,x)σ(x,y)=(y,x)^{\sigma}

for any x,yVx,y\in V. For any kk-subspace UVU\subseteq V, denote by

U={xV:(x,U)=0}U^{\perp}=\{x\in V:(x,U)=0\}

its orthogonal complement.

More generally, for any kk-scheme SS, put VS:=Vk𝒪SV_{S}:=V\otimes_{k}\mathcal{O}_{S}. Then there is a unique pairing (,)S:VS×VS𝒪S(\ ,\ )_{S}:V_{S}\times V_{S}\rightarrow\mathcal{O}_{S} extending (,)(\ ,\ ) that is 𝒪S\mathcal{O}_{S}-linear in the first variable and (𝒪S,σ)(\mathcal{O}_{S},\sigma)-linear in the second variable, given by

(xλ,yμ)=λμσ(x,y)(x\otimes\lambda,y\otimes\mu)=\lambda\mu^{\sigma}(x,y)

for any x,yVx,y\in V and λ,μ𝒪S\lambda,\mu\in\mathcal{O}_{S}. For any subbundle UVSU\subseteq V_{S}, we denote by

U={xVS:(x,U)=0}U^{\perp}=\{x\in V_{S}:(x,U)=0\}

its left orthogonal complement. (Notice that unlike the case S=SpeckS=\operatorname{Spec}k, in general the symmetry (2.0.1) does not necessarily hold for x,yVSx,y\in V_{S}, and the left orthogonal complement UU^{\perp} does not necessarily agree with the right orthogonal complement.)

2.2. Unitary Deligne–Lusztig varieties and special cycles

From now on fix a k/k0k/k_{0}-hermitian space VV of dimension n=2d+1n=2d+1 (d0d\geq 0).

Definition 2.1.

Define the unitary Deligne–Lusztig variety DL(V)Grd+1(V)\operatorname{DL}(V)\subseteq\operatorname{Gr}_{d+1}(V) to be the closed kk-subscheme parametrizing (d+1)(d+1)-dimensional subspaces UU such that UUU^{\perp}\subseteq U, i.e., for any kk-scheme SS,

DL(V)(S)={subbundles UVS:rankU=d+1,UU}.\operatorname{DL}(V)(S)=\{\text{subbundles }U\subseteq V_{S}:\operatorname{rank}U=d+1,\ U^{\perp}\subseteq U\}.

The relation between DL(V)\operatorname{DL}(V) and classical Deligne–Lusztig varieties associated to the unitary group U(V)\mathrm{U}(V) can be found in [Vol10, Corollary 2.17] and the reader can recognize that the Frobenius appearing in the definition of classical Deligne–Lusztig varieties arises from the σ\sigma-linearity in the second variable of the hermitian form (,)(\ ,\ ).

Definition 2.2.

Define a special subspace WVW\subseteq V to be a kk-subspace such that WWW^{\perp}\subseteq W. If WW is a special subspace, then W/WW/W^{\perp} is a (nondegenerate) k/k0k/k_{0}-hermitian space under the pairing induced from VV.

Definition 2.3.

Let WVW\subseteq V be a kk-subspace. Define the special cycle Z(W)DL(V)Z(W)\subseteq\operatorname{DL}(V) to be the closed subscheme parametrizing subspaces UU satisfying UWU\subseteq W, i.e., for any kk-scheme SS,

Z(W)(S)={UDL(V)(S):UWS}.Z(W)(S)=\{U\in\operatorname{DL}(V)(S):U\subseteq W_{S}\}.

By definition it is clear that

  • \bullet

    Z(W)Z(W) is nonempty only if WVW\subseteq V is a special subspace.

  • \bullet

    Z(V)=DL(V)Z(V)=\operatorname{DL}(V),

  • \bullet

    Z(W)Z(W)=Z(WW)Z(W)\cap Z(W^{\prime})=Z(W\cap W^{\prime}) for two subspaces W,WVW,W^{\prime}\subseteq V (\cap denotes the scheme-theoretic intersection).

  • \bullet

    Z(W)Z(W)Z(W)\subseteq Z(W^{\prime}) if WWW\subseteq W^{\prime}.

We summarize several known results on DL(V)\operatorname{DL}(V) and Z(W)Z(W) which we will need later.

Proposition 2.4.
  • (i)

    DL(V)\operatorname{DL}(V) is smooth, projective and geometrically irreducible of dimension dd.

  • (ii)

    Let WVW\subseteq V be a special subspace of codimension rr. Then there is a canonical isomorphism of kk-schemes

    (2.4.1) Z(W)DL(W/W).Z(W)\simeq\operatorname{DL}(W/W^{\perp}).

    In particular, Z(W)Z(W) is smooth, projective and geometrically irreducible, and has codimension rr in DL(V)\operatorname{DL}(V), and we call Z(W)Z(W) a codimension rr special cycle in DL(V)\operatorname{DL}(V).

  • (iii)

    The Tate conjecture (see §1.4) holds for DL(V)\operatorname{DL}(V).

  • (iv)

    For any r0r\geq 0, the space Tate2r(DL(V)k¯)\mathrm{Tate}_{\ell}^{2r}(\operatorname{DL}(V)_{{\bar{k}}}) is spanned by the cycle classes of codimension rr special cycles [Z(W)][Z(W)], where WVW\subseteq V runs over all special subspace of codimension rr.

  • (v)

    For any r0r\geq 0, the cup product induces a perfect pairing

    Tate2r(DL(V)k¯)×Tate2d2r(DL(V)k¯).\mathrm{Tate}_{\ell}^{2r}(\operatorname{DL}(V)_{{\bar{k}}})\times\mathrm{Tate}_{\ell}^{2d-2r}(\operatorname{DL}(V)_{{\bar{k}}})\xrightarrow{\cdot}\mathbb{Q}_{\ell}.
Proof.
  • (i)

    This is [VW11, Lemma 4.5], or the more general [LTX+19, Proposition A.1.3 (2)].

  • (ii)

    Consider the morphism Z(W)DL(W/W)Z(W)\rightarrow\operatorname{DL}(W/W^{\perp}) defined by

    Z(W)(S)DL(W/W)(S),UU¯=U/WSZ(W)(S)\rightarrow\operatorname{DL}(W/W^{\perp})(S),\quad U\overline{U}=U/W^{\perp}_{S}

    for any kk-scheme SS. It is an isomorphism, with inverse given by the morphism defined by U¯WS+U¯\overline{U}W^{\perp}_{S}+\overline{U}. The rest follows from Item (i) since W/WW/W^{\perp} has kk-dimension n2rn-2r.

  • (iii)

    This is [LZ22a, Theorem 5.3.2 (i)].

  • (iv)

    This is a combination of [LZ22a, Theorem 5.3.2 (i)] and [Vol10, Corollary 2.17].

  • (v)

    This is [LZ22a, (6.4.0.4)]. ∎

2.3. Natural vector bundles on DL(V)\operatorname{DL}(V) and Z(W)Z(W)

Definition 2.5.

Define

  • \bullet

    𝒱DL(V):=Vk𝒪DL(V)\mathcal{V}_{\operatorname{DL}(V)}:=V\otimes_{k}\mathcal{O}_{\operatorname{DL}(V)} the universal bundle on DL(V)\operatorname{DL}(V),

  • \bullet

    𝒰DL(V)𝒱DL(V)\mathcal{U}_{\operatorname{DL}(V)}\subseteq\mathcal{V}_{\operatorname{DL}(V)} and 𝒰DL(V)𝒱DL(V)\mathcal{U}_{\operatorname{DL}(V)}^{\perp}\subseteq\mathcal{V}_{\operatorname{DL}(V)} the two universal subbundles on DL(V)\operatorname{DL}(V),

  • \bullet

    𝒬DL(V):=𝒱DL(V)/𝒰DL(V)\mathcal{Q}_{\operatorname{DL}(V)}:=\mathcal{V}_{\operatorname{DL}(V)}/\mathcal{U}_{\operatorname{DL}(V)} the universal quotient bundle on DL(V)\operatorname{DL}(V),

  • \bullet

    DL(V):=𝒰DL(V)/𝒰DL(V),\mathcal{L}_{\operatorname{DL}(V)}:=\mathcal{U}_{\operatorname{DL}(V)}/\mathcal{U}_{\operatorname{DL}(V)}^{\perp}, which we call the special line bundle on DL(V)\operatorname{DL}(V).

Similarly for a special cycle Z(W)DL(V)Z(W)\subseteq\operatorname{DL}(V) associated to a special subspace WVW\subseteq V, define

  • \bullet

    𝒲Z(W):=(W/W)k𝒪Z(W)\mathcal{W}_{Z(W)}:=(W/W^{\perp})\otimes_{k}\mathcal{O}_{Z(W)} the universal bundle on Z(W)DL(W/W)Z(W)\simeq\operatorname{DL}(W/W^{\perp}),

  • \bullet

    𝒰Z(W)𝒲Z(W)\mathcal{U}_{Z(W)}\subseteq\mathcal{W}_{Z(W)} and 𝒰Z(W)𝒲Z(W)\mathcal{U}_{Z(W)}^{\perp}\subseteq\mathcal{W}_{Z(W)} the two universal subbundles on Z(W)Z(W),

  • \bullet

    𝒬Z(W):=𝒲Z(W)/𝒰Z(W)\mathcal{Q}_{Z(W)}:=\mathcal{W}_{Z(W)}/\mathcal{U}_{Z(W)} the universal quotient bundle on Z(W)Z(W),

  • \bullet

    Z(W):=𝒰Z(W)/𝒰Z(W),\mathcal{L}_{Z(W)}:=\mathcal{U}_{Z(W)}/\mathcal{U}_{Z(W)}^{\perp}, which we call the special line bundle on Z(W)Z(W).

Denote by

jW:Z(W)DL(V)j_{W}:Z(W)\hookrightarrow\operatorname{DL}(V)

the natural closed embedding. By definition, we have

(2.5.1) Z(W)=jWDL(V).\mathcal{L}_{Z(W)}=j_{W}^{*}\mathcal{L}_{\operatorname{DL}(V)}.

Also by definition, the canonical isomorphism (2.4.1) induces a canonical isomorphism Z(W)DL(W/W)\mathcal{L}_{Z(W)}\simeq\mathcal{L}_{\operatorname{DL}(W/W^{\perp})} and thus

(2.5.2) jWDL(V)DL(W/W).j_{W}^{*}\mathcal{L}_{\operatorname{DL}(V)}\simeq\mathcal{L}_{\operatorname{DL}(W/W^{\perp})}.
Proposition 2.6.

Let WVW\subseteq V be a special subspace of codimension rr.

  • (i)

    We have a canonical isomorphism

    (2.6.1) jW𝒬DL(V)/𝒬Z(W)(V/W)k𝒪Z(W).j_{W}^{*}\mathcal{Q}_{\operatorname{DL}(V)}/\mathcal{Q}_{Z(W)}\simeq(V/W)\otimes_{k}\mathcal{O}_{Z(W)}.
  • (ii)

    Let 𝒩Z(W)/DL(V)\mathcal{N}_{Z(W)/\operatorname{DL}(V)} be the normal bundle of Z(W)Z(W) in DL(V)\operatorname{DL}(V). Then we have an isomorphism

    𝒩Z(W)/DL(V)(Z(W)r).\mathcal{N}_{Z(W)/\operatorname{DL}(V)}\simeq\left(\mathcal{L}_{Z(W)}^{\oplus r}\right)^{*}.

    In particular, when WVW\subseteq V has codimension 1, we have an isomorphism of line bundles

    (2.6.2) 𝒩Z(W)/DL(V)Z(W).\mathcal{N}_{Z(W)/\operatorname{DL}(V)}\simeq\mathcal{L}_{Z(W)}^{*}.
Proof.
  • (i)

    The result follows immediately from the definition of 𝒬DL(V)\mathcal{Q}_{\operatorname{DL}(V)} and 𝒬Z(W)\mathcal{Q}_{Z(W)}.

  • (ii)

    By [LTX+19, Proposition A.1.3 (2)], we have canonical isomorphisms for the tangent bundles

    𝒯DL(V)/kom(DL(V),𝒬DL(V)),𝒯Z(W)/kom(Z(W),𝒬Z(W)).\mathcal{T}_{\operatorname{DL}(V)/k}\simeq\mathcal{H}om(\mathcal{L}_{\operatorname{DL}(V)},\mathcal{Q}_{\operatorname{DL}(V)}),\quad\mathcal{T}_{Z(W)/k}\simeq\mathcal{H}om(\mathcal{L}_{Z(W)},\mathcal{Q}_{Z(W)}).

    Hence by the exact sequence

    0𝒯Z(W)/kjW𝒯DL(V)/k𝒩Z(W)/DL(V)00\rightarrow\mathcal{T}_{Z(W)/k}\rightarrow j_{W}^{*}\mathcal{T}_{\operatorname{DL}(V)/k}\rightarrow\mathcal{N}_{Z(W)/\operatorname{DL}(V)}\rightarrow 0

    and (2.5.1), we obtain a canonical isomorphism

    𝒩Z(W)/DL(V)om(Z(W),jW𝒬DL(V)/𝒬Z(W)).\mathcal{N}_{Z(W)/\operatorname{DL}(V)}\simeq\mathcal{H}om(\mathcal{L}_{Z(W)},j_{W}^{*}\mathcal{Q}_{\operatorname{DL}(V)}/\mathcal{Q}_{Z(W)}).

    The result then follows from (2.6.1) as (V/W)k𝒪Z(W)𝒪Z(W)r(V/W)\otimes_{k}\mathcal{O}_{Z(W)}\simeq\mathcal{O}_{Z(W)}^{r}. ∎

2.4. Relations with Grassmannians

For 0mn0\leq m\leq n, let Grm(V)\operatorname{Gr}_{m}(V) be the Grassmannian of mm-dimensional subspaces of VV, which is a smooth projective variety over kk of dimension m(nm)m(n-m). We have the Plücker embedding

Pl:Grm(V)(m(V))N,N=(nm)1,\operatorname{Pl}:\operatorname{Gr}_{m}(V)\hookrightarrow\mathbb{P}(\wedge^{m}(V))\simeq\mathbb{P}^{N},\quad N=\textstyle{n\choose m}-1,

defined by sending an mm-dimensional subspace with basis {e1,,em}\{e_{1},\ldots,e_{m}\} to the line generated by e1emm(V)e_{1}\wedge\cdots\wedge e_{m}\in\wedge^{m}(V) (independent of the choice of the basis).

Definition 2.7.

Define 𝒮Grd+1(V)\mathcal{S}_{\operatorname{Gr}_{d+1}(V)} (resp. 𝒬Grd+1(V)\mathcal{Q}_{\operatorname{Gr}_{d+1}(V)}) to be the universal subbundle (resp. universal quotient subbundle) on Grd+1(V)\operatorname{Gr}_{d+1}(V). Denote by

i:DL(V)Grd+1(V)i:\operatorname{DL}(V)\hookrightarrow\operatorname{Gr}_{d+1}(V)

the natural closed embedding.

Lemma 2.8.

The following identities holds:

  • (i)

    𝒰DL(V)=i𝒮Grd+1(V)\mathcal{U}_{\operatorname{DL}(V)}=i^{*}\mathcal{S}_{\operatorname{Gr}_{d+1}(V)},

  • (ii)

    𝒰DL(V)i(σ𝒬Grd+1(V))\mathcal{U}_{\operatorname{DL}(V)}^{\perp}\simeq i^{*}(\sigma^{*}\mathcal{Q}_{\operatorname{Gr}_{d+1}(V)})^{*},

  • (iii)

    c1(𝒮Grd+1(V))=c1(𝒬Grd+1(V))c_{1}(\mathcal{S}_{\operatorname{Gr}_{d+1}(V)})=-c_{1}(\mathcal{Q}_{\operatorname{Gr}_{d+1}(V)}),

  • (iv)

    c1(DL(V))=(1q)ic1(𝒮Grd+1(V))c_{1}(\mathcal{L}_{\operatorname{DL}(V)})=(1-q)i^{*}c_{1}(\mathcal{S}_{\operatorname{Gr}_{d+1}(V)}).

Proof.
  • (i)

    It follows from the definition of 𝒮Grd+1(V)\mathcal{S}_{\operatorname{Gr}_{d+1}(V)} and 𝒰DL(V)\mathcal{U}_{\operatorname{DL}(V)}.

  • (ii)

    It follows from Item (i) and the nondegenerate pairing (,)(\ ,\ ) (σ\sigma-linear in the second variable).

  • (iii)

    It follows from the defining exact sequence

    0𝒮Grd+1(V)(Vk𝒪Grd+1(V))𝒬Grd+1(V)0.0\rightarrow\mathcal{S}_{\operatorname{Gr}_{d+1}(V)}\rightarrow(V\otimes_{k}\mathcal{O}_{\operatorname{Gr}_{d+1}(V)})\rightarrow\mathcal{Q}_{\operatorname{Gr}_{d+1}(V)}\rightarrow 0.
  • (iv)

    By the definition of DL(V)\mathcal{L}_{\operatorname{DL}(V)} together with Item (i) and Item (ii), we have

    c1(DL(V))=c1(𝒰DL(V))c1(𝒰DL(V))=ic1(𝒮Grd+1(V))ic1((σ𝒬Grd+1(V))).c_{1}(\mathcal{L}_{\operatorname{DL}(V)})=c_{1}(\mathcal{U}_{\operatorname{DL}(V)})-c_{1}(\mathcal{U}_{\operatorname{DL}(V)}^{\perp})=i^{*}c_{1}(\mathcal{S}_{\operatorname{Gr}_{d+1}(V)})-i^{*}c_{1}((\sigma^{*}\mathcal{Q}_{\operatorname{Gr}_{d+1}(V)})^{*}).

    By Item (iii), this evaluates to

    ic1(𝒮Grd+1(V))q(ic1(𝒮Grd+1(V)))=(1q)ic1(𝒮Grd+1(V)),i^{*}c_{1}(\mathcal{S}_{\operatorname{Gr}_{d+1}(V)})-q(i^{*}c_{1}(\mathcal{S}_{\operatorname{Gr}_{d+1}(V)}))=(1-q)i^{*}c_{1}(\mathcal{S}_{\operatorname{Gr}_{d+1}(V)}),

    as desired. ∎

Example 2.9.

When d=1d=1 (i.e., when dimV=3\dim V=3), we have Grd+1(V)2\operatorname{Gr}_{d+1}(V)\simeq\mathbb{P}^{2} with 𝒬Grd+1(V)𝒪2(1)\mathcal{Q}_{\operatorname{Gr}_{d+1}(V)}\simeq\mathcal{O}_{\mathbb{P}^{2}}(1). Thus c1(𝒮Grd+1(V))=c1(𝒪2(1))c_{1}(\mathcal{S}_{\operatorname{Gr}_{d+1}(V)})=c_{1}(\mathcal{O}_{\mathbb{P}^{2}}(-1)) by Lemma 2.8 (iii). By definition DL(V)Grd+1(V)\operatorname{DL}(V)\subseteq\operatorname{Gr}_{d+1}(V) is isomorphic to a Fermat curve of degree 1+q1+q ([Vol10, Remark 4.7]),

{[x,y,z]2:xq+1+yq+1+zq+1=0}2.\{[x,y,z]\in\mathbb{P}^{2}:x^{q+1}+y^{q+1}+z^{q+1}=0\}\subseteq\mathbb{P}^{2}.

Hence by Lemma 2.8 (iv), the following identity holds in Ch1(DL(V)k¯){\mathrm{Ch}}^{1}(\operatorname{DL}(V)_{{\bar{k}}})_{\mathbb{Q}}\simeq\mathbb{Q},

c1(DL(V))=(1q)ic1(𝒪2(1))=(1q)(1+q)=(1q2),c_{1}(\mathcal{L}_{\operatorname{DL}(V)})=(1-q)i^{*}c_{1}(\mathcal{O}_{\mathbb{P}^{2}}(-1))=-(1-q)(1+q)=-(1-q^{2}),

and so

(2.9.1) c1(DL(V))=c1(DL(V))=1q2.c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})=-c_{1}(\mathcal{L}_{\operatorname{DL}(V)})=1-q^{2}.

2.5. Chern classes of the special line bundle

Lemma 2.10.

Let Z(W)DL(V)Z(W)\subseteq\operatorname{DL}(V) be a 1-dimensional special cycle associated to a special subspace WVW\subseteq V of codimension d1d-1. Then the following identity holds in Chd(DL(V)k¯){\mathrm{Ch}}^{d}(\operatorname{DL}(V)_{{\bar{k}}})_{\mathbb{Q}}\simeq\mathbb{Q}:

c1(DL(V))[Z(W)]=1q2.c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})\cdot[Z(W)]=1-q^{2}.
Proof.

By the projection formula we know that

c1(DL(V))[Z(W)]=jW,(c1(jWDL(V))).c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})\cdot[Z(W)]=j_{W,*}(c_{1}(j_{W}^{*}\mathcal{L}_{\operatorname{DL}(V)})).

By (2.5.2), we have (under the isomorphism (2.4.1))

c1(jWDL(V))=c1(DL(W/W)).c_{1}(j_{W}^{*}\mathcal{L}_{\operatorname{DL}(V)}^{*})=c_{1}(\mathcal{L}_{\operatorname{DL}(W/W^{\perp})}^{*}).

As W/WW/W^{\perp} is of dimension 3, we know the latter evaluates to 1q21-q^{2} in Ch1(DL(W/W)k¯){\mathrm{Ch}}^{1}(\operatorname{DL}(W/W^{\perp})_{\bar{k}})_{\mathbb{Q}}\simeq\mathbb{Q} by (2.9.1). The result then follows. ∎

Proposition 2.11.

The following identity holds in Ch1(DL(V)k¯){\mathrm{Ch}}^{1}(\operatorname{DL}(V)_{{\bar{k}}})_{\mathbb{Q}}:

(2.11.1) c1(DL(V))=1q21+q2d+1codimW=1[Z(W)],c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})=\frac{1-q^{2}}{1+q^{2d+1}}\sum_{\operatorname{codim}W=1}[Z(W)],

here the sum runs over all special subspaces WVW\subseteq V of codimension 1.

Proof.

When d=1d=1, the number of special subspaces WVW\subseteq V is equal to the number of isotropic lines in the 3-dimensional k/k0k/k_{0}-hermitian space VV, which is 1+q31+q^{3} (e.g., by [LZ22a, Lemma 1.9.1]). Hence the right hand side of (2.11.1) evaluates to

1q21+q3(1+q3)=1q2\frac{1-q^{2}}{1+q^{3}}\cdot(1+q^{3})=1-q^{2}

in Ch1(DL(V)k¯){\mathrm{Ch}}^{1}(\operatorname{DL}(V)_{{\bar{k}}})_{\mathbb{Q}}\simeq\mathbb{Q}, which agrees with the left hand side by (2.9.1).

When d>1d>1, by Proposition 2.4 (iv) (v) and the commutativity of (1.4.1), to prove (2.11.1) it suffices to show that for any 1-dimensional special cycle Z(W)DL(V)Z(W^{\prime})\subseteq\operatorname{DL}(V) (associated to any codimension d1d-1 special subspace WVW^{\prime}\subseteq V), the following identity holds in Chd(DL(V)k¯){\mathrm{Ch}}^{d}(\operatorname{DL}(V)_{{\bar{k}}})_{\mathbb{Q}}\simeq\mathbb{Q},

(2.11.2) c1(DL(V))[Z(W)]=1q21+q2d+1codimW=1[Z(W)][Z(W)].c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})\cdot[Z(W^{\prime})]=\frac{1-q^{2}}{1+q^{2d+1}}\sum_{\operatorname{codim}W=1}[Z(W)]\cdot[Z(W^{\prime})].

For the terms of the right hand side of (2.11.2), we have three cases.

  • (i)

    When WWW^{\prime}\subseteq W, we have Z(W)Z(W)Z(W^{\prime})\subseteq Z(W). By the excess intersection formula [Ful98, Corollary 6.3], we know that

    [Z(W)][Z(W)]=jW,(c1(𝒩Z(W)/DL(V))[Z(W)]Z(W)).[Z(W)]\cdot[Z(W^{\prime})]=j_{W,*}(c_{1}(\mathcal{N}_{Z(W)/\operatorname{DL}(V)})\cdot[Z(W^{\prime})]_{Z(W)}).

    By (2.6.2) and (2.5.2) we have (under the isomorphism (2.4.1))

    c1(𝒩Z(W)/DL(V))[Z(W)]Z(W)=c1(DL(W/W))[Z(W)]DL(W/W),c_{1}(\mathcal{N}_{Z(W)/\operatorname{DL}(V)})\cdot[Z(W^{\prime})]_{Z(W)}=c_{1}(\mathcal{L}_{\operatorname{DL}(W/W^{\perp})}^{*})\cdot[Z(W^{\prime})]_{\operatorname{DL}(W/W^{\perp})},

    which evaluates to 1q21-q^{2} in Chd1(DL(W/W)k¯){\mathrm{Ch}}^{d-1}(\operatorname{DL}(W/W^{\perp})_{\bar{k}})_{\mathbb{Q}}\simeq\mathbb{Q} by Lemma 2.10 applied to the 1-dimensional special cycle Z(W)DL(W/W)Z(W^{\prime})\subseteq\operatorname{DL}(W/W^{\perp}). Thus in this case

    [Z(W)][Z(W)]=1q2.[Z(W)]\cdot[Z(W^{\prime})]=1-q^{2}.
  • (ii)

    When WWW^{\prime}\not\subseteq W and WWVW^{\prime}\cap W\subseteq V is a special subspace, we know that WWW^{\prime}\cap W has codimension dd in VV and thus scheme-theoretic intersection Z(W)Z(W)=Z(WW)Z(W)\cap Z(W^{\prime})=Z(W^{\prime}\cap W) is isomorphic to Speck\operatorname{Spec}k by Proposition 2.4 (i), and thus

    [Z(W)][Z(W)]=1.[Z(W)]\cdot[Z(W^{\prime})]=1.
  • (iii)

    When WWW^{\prime}\not\subseteq W and WWVW^{\prime}\cap W\subseteq V is not a special subspace, we know that Z(W)Z(W)=Z(WW)Z(W)\cap Z(W^{\prime})=Z(W^{\prime}\cap W) is empty, and thus

    [Z(W)][Z(W)]=0.[Z(W)]\cdot[Z(W^{\prime})]=0.

Now we count the number of terms on the right hand side of (2.11.2) in first two cases.

  • \bullet

    The association WWWW^{\perp} gives a bijection between the set of codimension 1 special subspaces WVW\subseteq V in Case (i) and the set of isotropic kk-lines in WW^{\prime\perp}. Hence the number of terms in Case (i) is equal to 1q2(d1)1q2\frac{1-q^{2(d-1)}}{1-q^{2}}, as WW^{\prime\perp} is a totally isotropic k/k0k/k_{0}-hermitian space of dimension d1d-1.

  • \bullet

    The association WWWW^{\perp} gives a bijection between the set of codimension 1 special subspaces WVW\subseteq V in Case (ii) and the set of isotropic kk-lines in WWW^{\prime}\setminus W^{\prime\perp}. Hence the number of terms in Case (ii) is equal to the number of vectors in WW^{\prime\perp} times the number of isotropic lines in the k/k0k/k_{0}-hermitian space W/WW^{\prime}/W^{\prime\perp}. This is q2(d1)(1+q3)q^{2(d-1)}(1+q^{3}), as WW^{\prime\perp} is of dimension d1d-1 and W/WW^{\prime}/W^{\prime\perp} is of dimension 3.

Thus the right hand side of (2.11.2) evaluates to

1q21+q2d+1((1q2)1q2(d1)1q2+1q2(d1)(1+q3))=1q2,\frac{1-q^{2}}{1+q^{2d+1}}\cdot\left((1-q^{2})\cdot\frac{1-q^{2(d-1)}}{1-q^{2}}+1\cdot q^{2(d-1)}(1+q^{3})\right)=1-q^{2},

which is equal to the left hand side by Lemma 2.10 applied to the 1-dimensional special cycle Z(W)DL(V)Z(W^{\prime})\subseteq\operatorname{DL}(V). ∎

Proposition 2.12.

The following identity holds in Chd(DL(V)k¯){\mathrm{Ch}}^{d}(\operatorname{DL}(V)_{{\bar{k}}})_{\mathbb{Q}}\simeq\mathbb{Q}:

c1(DL(V))d=i=1d(1q2i).c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})^{d}=\prod_{i=1}^{d}(1-q^{2i}).
Proof.

When d=1d=1, this is (2.9.1). In general, we induct on dd. By Proposition 2.11, we obtain

(2.12.1) c1(DL(V))d=1q21+q2d+1c1(DL(V))d1codimW=1[Z(W)].c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})^{d}=\frac{1-q^{2}}{1+q^{2d+1}}\cdot c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})^{d-1}\cdot\sum_{\operatorname{codim}W=1}[Z(W)].

By the projection formula and (2.5.2), we have (under the isomorphism (2.4.1)),

c1(DL(V))d1[Z(W)]=jW,(c1(DL(W/W))d1),c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})^{d-1}\cdot[Z(W)]=j_{W,*}(c_{1}(\mathcal{L}_{\operatorname{DL}(W/W^{\perp})}^{*})^{d-1}),

which evaluates to i=1d1(1q2i)\prod_{i=1}^{d-1}(1-q^{2i}) by the induction hypothesis.

The association WWWW^{\perp} gives a bijection the set of codimension 1 special subspaces WVW\subseteq V and the set of isotropic kk-lines in VV. Thus the total number of terms in (2.12.1) is the number of isotropic kk-lines in the (2d+1)(2d+1)-dimensional k/k0k/k_{0}-hermitian space VV, which is (1+q2d+1)(1q2d)1q2\frac{(1+q^{2d+1})(1-q^{2d})}{1-q^{2}} (e.g., by [LZ22a, Lemma 1.9.1]). Thus (2.12.1) evaluates to

1q21+q2d+1i=1d1(1q2i)(1+q2d+1)(1q2d)1q2=i=1d(1q2i).\frac{1-q^{2}}{1+q^{2d+1}}\cdot\prod_{i=1}^{d-1}(1-q^{2i})\cdot\frac{(1+q^{2d+1})(1-q^{2d})}{1-q^{2}}=\prod_{i=1}^{d}(1-q^{2i}).

This completes the proof. ∎

2.6. Proof of Theorem 1.1

Recall Pl:Grd+1(V)(d+1V)\operatorname{Pl}:\operatorname{Gr}_{d+1}(V)\rightarrow\mathbb{P}(\wedge^{d+1}V) is the Plücker embedding. By definition

(2.12.2) degDL(V)=[DL(V)](d+1V)c1(𝒪(d+1V)(1))d\deg\operatorname{DL}(V)=[\operatorname{DL}(V)]_{\mathbb{P}(\wedge^{d+1}V)}\cdot c_{1}(\mathcal{O}_{\mathbb{P}(\wedge^{d+1}V)}(1))^{d}

is the intersection number of DL(V)\operatorname{DL}(V) with dd general hyperplanes in (d+1V)\mathbb{P}(\wedge^{d+1}V). By the projection formula, we obtain that

degDL(V)=c1((Pli)𝒪(d+1V)(1))d\deg\operatorname{DL}(V)=c_{1}((\operatorname{Pl}\circ i)^{*}\mathcal{O}_{\mathbb{P}(\wedge^{d+1}V)}(1))^{d}

in Chd(DL(V)k¯){\mathrm{Ch}}^{d}(\operatorname{DL}(V)_{\bar{k}})_{\mathbb{Q}}\simeq\mathbb{Q}. By the definition of the Plücker embedding, we have

Pl𝒪(d+1V)(1)det𝒮Grd+1(V).\operatorname{Pl}^{*}\mathcal{O}_{\mathbb{P}(\wedge^{d+1}V)}(1)\simeq\det\mathcal{S}_{\operatorname{Gr}_{d+1}(V)}^{*}.

Thus

c1((Pli)𝒪(d+1V)(1))=ic1(det𝒮Grd+1(V))=ic1(𝒮Grd+1(V)).c_{1}((\operatorname{Pl}\circ i)^{*}\mathcal{O}_{\mathbb{P}(\wedge^{d+1}V)}(1))=i^{*}c_{1}(\det\mathcal{S}_{\operatorname{Gr}_{d+1}(V)}^{*})=i^{*}c_{1}(\mathcal{S}_{\operatorname{Gr}_{d+1}(V)}^{*}).

By Lemma 2.8 (iv), we have

ic1(𝒮Grd+1(V))=c1(DL(V))1q,i^{*}c_{1}(\mathcal{S}_{\operatorname{Gr}_{d+1}(V)}^{*})=\frac{c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})}{1-q},

and hence

degDL(V)=c1(DL(V))d(1q)d.\deg\operatorname{DL}(V)=\frac{c_{1}(\mathcal{L}_{\operatorname{DL}(V)}^{*})^{d}}{(1-q)^{d}}.

The result then follows from Proposition 2.12.

3. Schubert calculus

3.1. Reminder on Schubert calculus (cf. [EH16, Chapter 4],[Ful98, §14.7])

Let 0mn0\leq m\leq n. The Schubert classes of Grm(V)\operatorname{Gr}_{m}(V) are indexed by mm-tuples a=(a1,,am)a=(a_{1},\ldots,a_{m}) of integers satisfying

nma1a2am0,n-m\geq a_{1}\geq a_{2}\geq\cdots\geq a_{m}\geq 0,

in other words, indexed by Young diagrams inside the m×(nm)m\times(n-m) rectangles. For such an mm-tuple aa, define a Schubert cycle

Σa(V):={UGrm(V):dimUVnm+iaii,i=1,,m}Grm(V),\Sigma_{a}(V_{\bullet}):=\{U\in\operatorname{Gr}_{m}(V):\dim U\cap V_{n-m+i-a_{i}}\geq i,\ i=1,\ldots,m\}\subseteq\operatorname{Gr}_{m}(V),

where

V:0V1V2Vn1Vn:=VV_{\bullet}:0\subset V_{1}\subset V_{2}\subset\cdots\subset V_{n-1}\subset V_{n}:=V

is a complete flag in VV. The Schubert cycle Σa(V)Grm(V)\Sigma_{a}(V_{\bullet})\subseteq\operatorname{Gr}_{m}(V) is a closed subvariety of codimension |a|:=i=1mai|a|:=\sum_{i=1}^{m}a_{i}. Define the Schubert class

σa:=[Σa(V)]Ch|a|(Grm(V)k¯),\sigma_{a}:=[\Sigma_{a}(V_{\bullet})]\in{\mathrm{Ch}}^{|a|}(\operatorname{Gr}_{m}(V)_{\bar{k}}),

which is independent of the choice of the complete flag VV_{\bullet}. We use the standard notation suppressing trailing zeros in the indices. In particular, by definition σ1=σ1,0,,0Ch1(Grm(V)k¯)\sigma_{1}=\sigma_{1,0,\ldots,0}\in{\mathrm{Ch}}^{1}(\operatorname{Gr}_{m}(V)_{\bar{k}}) is the hyperplane class under the Plücker embedding.

Define the dual a:=(nmam,,nma1)a^{*}:=(n-m-a_{m},\cdots,n-m-a_{1}). The Schubert classes form a \mathbb{Q}-basis of Ch(Grm(V)k¯){\mathrm{Ch}}^{*}(\operatorname{Gr}_{m}(V)_{{\bar{k}}})_{\mathbb{Q}} and the intersection pairing

Chr(Grm(V)k¯)×Chm(nm)r(Grm(V)k¯){\mathrm{Ch}}^{r}(\operatorname{Gr}_{m}(V)_{{\bar{k}}})_{\mathbb{Q}}\times{\mathrm{Ch}}^{m(n-m)-r}(\operatorname{Gr}_{m}(V)_{{\bar{k}}})_{\mathbb{Q}}\rightarrow\mathbb{Q}

is perfect and has Schubert classes as dual basis, with σa\sigma_{a} and σb\sigma_{b} dual to each other if and only if b=ab=a^{*}.

Define the conjugate a:=(a1,,anm)a^{\prime}:=(a^{\prime}_{1},\ldots,a^{\prime}_{n-m}) such that aja^{\prime}_{j} is the number of ii’s such that aija_{i}\geq j . The Young diagrams of aa and aa^{\prime} are mutual reflections along the main diagonal. The canonical isomorphism

(3.0.1) Grm(V)Grnm(V),U(V/U)\operatorname{Gr}_{m}(V)\simeq\operatorname{Gr}_{n-m}(V^{*}),\quad U(V/U)^{*}

maps σaCh|a|(Grm(V))\sigma_{a}\in{\mathrm{Ch}}^{|a|}(\operatorname{Gr}_{m}(V)) to σaCh|a|(Grnm(V))\sigma_{a^{\prime}}\in{\mathrm{Ch}}^{|a|}(\operatorname{Gr}_{n-m}(V^{*})).

3.2. The class of DL(V)\operatorname{DL}(V)

Let Grd,d+1(V)\operatorname{Gr}_{d,d+1}(V) be the variety parametrizing partial flags

U:0UdUd+1V,U_{\bullet}:0\subseteq U_{d}\subseteq U_{d+1}\subseteq V,

where dimUd=d\dim U_{d}=d and dimUd+1=d+1\dim U_{d+1}=d+1. Define a closed embedding

ψ:Grd,d+1(V)Grd(V)×Grd+1(V),U(Ud,Ud+1).\psi:\operatorname{Gr}_{d,d+1}(V)\rightarrow\operatorname{Gr}_{d}(V)\times\operatorname{Gr}_{d+1}(V),\quad U_{\bullet}(U_{d},U_{d+1}).

Also consider the closed embedding

(ϕ,id):Grd+1(V)Grd(V)×Grd+1(V),U(U,U).(\phi,\mathrm{id}):\operatorname{Gr}_{d+1}(V)\rightarrow\operatorname{Gr}_{d}(V)\times\operatorname{Gr}_{d+1}(V),\quad U(U^{\perp},U).

Then by definition (ϕ,id)(\phi,\mathrm{id}) induces an isomorphism between DL(V)\operatorname{DL}(V) and im(ϕ,id)im(ψ)\operatorname{im}(\phi,\mathrm{id})\cap\operatorname{im}(\psi).

Proposition 3.1.

The following identity holds in Ch(Grd(V)k¯×Grd+1(V)k¯){\mathrm{Ch}}^{*}(\operatorname{Gr}_{d}(V)_{\bar{k}}\times\operatorname{Gr}_{d+1}(V)_{\bar{k}})_{\mathbb{Q}}:

[imψ]=a,bσa×σb,[\operatorname{im}\psi]=\sum_{a,b}\sigma_{a^{*}}\times\sigma_{b^{*}},

where the sum runs over

  • \bullet

    a=(a1,,ad)a=(a_{1},\ldots,a_{d}) satisfying d+1a1ad0d+1\geq a_{1}\geq\cdots\geq a_{d}\geq 0,

  • \bullet

    b=(b1,,bd+1)b=(b_{1},\ldots,b_{d+1}) satisfying b1=db_{1}=d, bi=d+1ad+2i0b_{i}=d+1-a_{d+2-i}\geq 0 for i=2,,d+1i=2,\ldots,d+1.

Proof.

Let VV_{\bullet} and WW_{\bullet} be two transverse complete flags in VV ([EH16, Definition 4.4]). Recall that VV_{\bullet} and WW_{\bullet} are transverse means that ViWni=0V_{i}\cap W_{n-i}=0 for any ii. Let

V(i):=Vnd+iai,W(i):=Wn(d+1)+ibi.V^{(i)}:=V_{n-d+i-a_{i}},\quad W^{(i)}:=W_{n-(d+1)+i-b_{i}}.

Then by definition

Σa(V)={UdGrd(V):dimUdV(i)i,i=1,,d}\Sigma_{a}(V_{\bullet})=\{U_{d}\in\operatorname{Gr}_{d}(V):\dim U_{d}\cap V^{(i)}\geq i,\quad i=1,\ldots,d\}

and

Σb(W)={Ud+1Grd+1(V):dimUd+1W(i)i,i=1,,d+1}.\Sigma_{b}(W_{\bullet})=\{U_{d+1}\in\operatorname{Gr}_{d+1}(V):\dim U_{d+1}\cap W^{(i)}\geq i,\quad i=1,\ldots,d+1\}.

By the transversality it is easy to see that the intersection of im(ψ)\operatorname{im}(\psi) with Σa(V)×Σb(W)Grd(V)×Grd+1(V)\Sigma_{a}(V_{\bullet})\times\Sigma_{b}(W_{\bullet})\subseteq\operatorname{Gr}_{d}(V)\times\operatorname{Gr}_{d+1}(V) is nonempty if and only if

dimW(1)=1,dimW(i)V(d+2i)=1,i=2,d+1,\dim W^{(1)}=1,\quad\dim W^{(i)}\cap V^{(d+2-i)}=1,\quad i=2,\ldots d+1,

in which case the intersection is transverse at the unique point given by

Ud=i=2d+1W(i)V(d+2i),Ud+1=UdW(1).U_{d}=\bigoplus_{i=2}^{d+1}W^{(i)}\cap V^{(d+2-i)},\quad U_{d+1}=U_{d}\oplus W^{(1)}.

Therefore

[imψ](σa×σb)={1,b1=d,bi=d+1ad+2i,i=2,,d+1,0,otherwise.[\operatorname{im}\psi]\cdot(\sigma_{a}\times\sigma_{b})=\begin{cases}1,&b_{1}=d,\quad b_{i}=d+1-a_{d+2-i},i=2,\ldots,d+1,\\ 0,&\text{otherwise}.\end{cases}

The desired result then follows from the fact that Schubert classes form dual basis under the intersection pairing. ∎

Corollary 3.2.

The following identity holds in Ch(Grd+1(V)k¯){\mathrm{Ch}}^{*}(\operatorname{Gr}_{d+1}(V)_{\bar{k}})_{\mathbb{Q}}:

[DL(V)]=cσcσc^q|c|,[\operatorname{DL}(V)]=\sum_{c}\sigma_{c}\sigma_{\widehat{c}^{\prime}}q^{|c|},

where the sum runs over c=(c1,,cd)c=(c_{1},\ldots,c_{d}) such that dc1cd0d\geq c_{1}\geq\cdots\geq c_{d}\geq 0, and c^\widehat{c} is the complement of cc defined by c^:=(dcd,,dc1)\hat{c}:=(d-c_{d},\ldots,d-c_{1}).

Proof.

By definition we have [DL(V)]=(ϕ,id)[im(ψ)][\operatorname{DL}(V)]=(\phi,\mathrm{id})^{*}[\operatorname{im}(\psi)]. Since Uσ(V/U)U^{\perp}\simeq\sigma^{*}(V/U)^{*}, by (3.0.1) it is easy to see that

(ϕ,id)(σa×σb)=q|a|σ(a)σbCh(Grd+1(V)k¯).(\phi,\mathrm{id})^{*}(\sigma_{a^{*}}\times\sigma_{b^{*}})=q^{|a^{*}|}\sigma_{(a^{*})^{\prime}}\sigma_{b^{*}}\in{\mathrm{Ch}}^{*}(\operatorname{Gr}_{d+1}(V)_{\bar{k}})_{\mathbb{Q}}.

Since b=(d,d+1ad,d+1a2,,d+1a1)b=(d,d+1-a_{d},d+1-a_{2},\ldots,d+1-a_{1}), we know that b=(a11,,ad1,0)b^{*}=(a_{1}-1,\ldots,a_{d}-1,0). Let c=(a11,,ad1)c=(a_{1}-1,\ldots,a_{d}-1). Then c^=(d+1ad,,d+1a1)=a\widehat{c}=(d+1-a_{d},\ldots,d+1-a_{1})=a^{*}. It follows from Proposition 3.1 that

[DL(V)]=cσc^σcq|c^|=cσcσc^q|c|.[\operatorname{DL}(V)]=\sum_{c}\sigma_{\widehat{c}^{\prime}}\sigma_{c}q^{|\widehat{c}|}=\sum_{c}\sigma_{c}\sigma_{\widehat{c}^{\prime}}q^{|c|}.

This completes the proof.∎

Example 3.3.
  • \bullet

    When d=1d=1, by Corollary 3.2 we obtain

    [DL(V)]=σ1+σ1q=(1+q)σ1.[\operatorname{DL}(V)]=\sigma_{1}+\sigma_{1}q=(1+q)\sigma_{1}.

    This agrees with the fact that DL(V)Grd+1(V)2\operatorname{DL}(V)\subseteq\operatorname{Gr}_{d+1}(V)\simeq\mathbb{P}^{2} is the Fermat curve of degree 1+q1+q (Example 2.9).

  • \bullet

    When d=2d=2, by Corollary 3.2 we obtain

    [DL(V)]=σ2,2+σ1σ2,1q+σ2σ1,1q2+σ1,1σ2q2+σ2,1σ1q3+σ2,2q4.[\operatorname{DL}(V)]=\sigma_{2,2}+\sigma_{1}\sigma_{2,1}q+\sigma_{2}\sigma_{1,1}q^{2}+\sigma_{1,1}\sigma_{2}q^{2}+\sigma_{2,1}\sigma_{1}q^{3}+\sigma_{2,2}q^{4}.

3.3. Proof of Theorem 1.3

By (2.12.2) and the projection formula, we have

degDL(V)=[DL(V)]c1(Pl𝒪(d+1V)(1))d.\deg\operatorname{DL}(V)=[\operatorname{DL}(V)]\cdot c_{1}(\operatorname{Pl}^{*}\mathcal{O}_{\mathbb{P}(\wedge^{d+1}V)}(1))^{d}.

The result then follows from Corollary 3.2 and σ1=c1(Pl𝒪(d+1V)(1))\sigma_{1}=c_{1}(\operatorname{Pl}^{*}\mathcal{O}_{\mathbb{P}(\wedge^{d+1}V)}(1)) is the hyperplane place class under the Plücker embedding.

3.4. Proof of Corollary 1.4

  • (i)

    This follows immediately from Theorems 1.1 and 1.3.

  • (ii)

    Let

    Sc(x1,,xd)=det(xicj+dj)/det(xidj)S_{c}(x_{1},\ldots,x_{d})=\det(x_{i}^{c_{j}+d-j})/\det(x_{i}^{d-j})

    (1i,jd1\leq i,j\leq d) be the Schur polynomial associated to c=(c1,,cd)c=(c_{1},\ldots,c_{d}) ([Mac92]). It is a symmetric polynomial of degree |c||c|. Write

    S:=cSc(qx1,,qxd)Sc^(x1,,xd)S1d(x1,,xd)=:λκλSλ,S:=\sum_{c}S_{c}(qx_{1},\ldots,qx_{d})S_{\widehat{c}^{\prime}}(x_{1},\ldots,x_{d})S_{1}^{d}(x_{1},\ldots,x_{d})=:\sum_{\lambda}\kappa_{\lambda}S_{\lambda},

    where λ\lambda runs over λ=(λ1,,λd)\lambda=(\lambda_{1},\ldots,\lambda_{d}) with |λ|=d(d+1)|\lambda|=d(d+1). By definition, the coefficient κλ\kappa_{\lambda} is given by the coefficient of x1λ1+d1x2λ2+d2xdλdx_{1}^{\lambda_{1}+d-1}x_{2}^{\lambda_{2}+d-2}\cdots x_{d}^{\lambda_{d}} in

    (3.3.1) Sdet(xidj)=Si<j(xixj).S\cdot\det(x_{i}^{d-j})=S\cdot\prod_{i<j}(x_{i}-x_{j}).

    Since the class of a point is σ(d+1,,d+1)\sigma_{(d+1,\ldots,d+1)}, we know that

    cσcσc^σ1dq|c|=κ(d+1,d+1,,d+1),\sum_{c}\sigma_{c}\sigma_{\widehat{c}^{\prime}}\sigma_{1}^{d}q^{|c|}=\kappa_{(d+1,d+1,\ldots,d+1)},

    i.e., the coefficient of x12dx22d1xdd+1x_{1}^{2d}x_{2}^{2d-1}\cdots x_{d}^{d+1} in (3.3.1). It remains to compute (3.3.1). By the dual Cauchy identity for Schur polynomials ([Mac92, 0.11’]), we have

    cSc(x1,,xd)Sc^(y1,,yd)=i,j=1d(xi+yj).\sum_{c}S_{c}(x_{1},\ldots,x_{d})S_{\widehat{c}^{\prime}}(y_{1},\ldots,y_{d})=\prod_{i,j=1}^{d}(x_{i}+y_{j}).

    By definition, we have

    S1(x1,,xd)=x1++xd.S_{1}(x_{1},\ldots,x_{d})=x_{1}+\cdots+x_{d}.

    Therefore

    S=cSc(qx1,,qxd)Sc^(x1,,xd)S1d(x1,,xd)=(i,j=1d(qxi+xj))(x1++xd)d,S=\sum_{c}S_{c}(qx_{1},\ldots,qx_{d})S_{\widehat{c}^{\prime}}(x_{1},\ldots,x_{d})S_{1}^{d}(x_{1},\ldots,x_{d})=\left(\prod_{i,j=1}^{d}(qx_{i}+x_{j})\right)(x_{1}+\cdots+x_{d})^{d},

    and thus

    (3.3.2) Si<j(xixj)=(i,j=1d(qxi+xj))(x1++xd)d(i<j(xixj)).S\cdot\prod_{i<j}(x_{i}-x_{j})=\left(\prod_{i,j=1}^{d}(qx_{i}+x_{j})\right)(x_{1}+\cdots+x_{d})^{d}\left(\prod_{i<j}(x_{i}-x_{j})\right).

    The result then follows from Item (i).

  • (iii)

    By applying Pieri’s formula ([EH16, Proposition 4.9]) dd times, we know that the term σcσc^σ1d\sigma_{c}\sigma_{\widehat{c}^{\prime}}\sigma_{1}^{d} is equal to the number of sequences of Young diagrams c(0),c(1),c(d)c^{(0)},c^{(1)}\ldots,c^{(d)} starting with c(0)=cc^{(0)}=c and ending with c(d)=(c^)c^{(d)}=(\widehat{c}^{\prime})^{*} such that each c(i+1)c^{(i+1)} has exactly one more box than c(i)c^{(i)}. Equivalently, it is the number of standard Young tableaux of skew shape (c^)/c(\widehat{c}^{\prime})^{*}/c. Now Item (i) shows that the number of such standard Young tableaux with |c|=l|c|=l is equal to the coefficient of qlq^{l} in

    i=1d1q2i1q=i=1dj=02i1qj,\prod_{i=1}^{d}{\frac{1-q^{2i}}{1-q}}=\prod_{i=1}^{d}\sum_{j=0}^{2i-1}q^{j},

    which is equal to the number of ordered partitions (l1,,ld)(l_{1},\ldots,l_{d}) of ll satisfying the extra conditions

    0li2i1,i=1,,d.0\leq l_{i}\leq 2i-1,\quad i=1,\ldots,d.
Example 3.4.

We end with an example illustrating Corollary 1.4 (ii) (iii).

  • \bullet

    When d=1d=1, the polynomial (3.3.2) is equal to (1+q)x12(1+q)x_{1}^{2}. The coefficient of x12x_{1}^{2} is given by 1+q=1q21q1+q=\frac{1-q^{2}}{1-q} as in Corollary 1.4 (ii).

  • \bullet

    When d=2d=2, the polynomial (3.3.2) is equal to

    (q+2q2+q3)x16x2+(1+3q+4q2+3q3+q4)x15x22+(1+2q+2q2+2q3+q4)x14x23\displaystyle\left(q+2q^{2}+q^{3}\right)x_{1}^{6}x_{2}+\left(1+3q+4q^{2}+3q^{3}+q^{4}\right)x_{1}^{5}x_{2}^{2}+\left(1+2q+2q^{2}+2q^{3}+q^{4}\right)x_{1}^{4}x_{2}^{3}
    +(12q2q22q3q4)x13x24+(13q4q23q3q4)x12x25+(q2q2q3)x1x26\displaystyle+\left(-1-2q-2q^{2}-2q^{3}-q^{4}\right)x_{1}^{3}x_{2}^{4}+\left(-1-3q-4q^{2}-3q^{3}-q^{4}\right)x_{1}^{2}x_{2}^{5}+\left(-q-2q^{2}-q^{3}\right)x_{1}x_{2}^{6}

    The coefficient of x14x23x_{1}^{4}x_{2}^{3} is given by

    1+2q+2q2+2q3+q4=(1+q)(1+q+q2+q3),1+2q+2q^{2}+2q^{3}+q^{4}=(1+q)(1+q+q^{2}+q^{3}),

    which equals 1q21q1q41q\frac{1-q^{2}}{1-q}\cdot\frac{1-q^{4}}{1-q} as in Corollary 1.4 (ii).

  • \bullet

    When d=3d=3, the coefficient of x16x25x34x_{1}^{6}x_{2}^{5}x_{3}^{4} is equal to

    1+3q+5q2+7q3+8q4+8q5+7q6+5q7+3q8+q9=(1+q)(1+q2+q3)(1+q+q2+q3+q4+q5),1+3q+5q^{2}+7q^{3}+8q^{4}+8q^{5}+7q^{6}+5q^{7}+3q^{8}+q^{9}=(1+q)(1+q^{2}+q^{3})(1+q+q^{2}+q^{3}+q^{4}+q^{5}),

    as in Corollary 1.4 (ii). In Table \bullet3.4 we list all standard Young tableaux of skew shape (c^)/c(\widehat{c}^{\prime})^{*}/c with |c|=4|c|=4 as in Corollary 1.4 (iii). Notice the total number of such Young tableaux is 8, which indeed agrees with the coefficient of q4q^{4}.

    cc (c^)(\widehat{c}^{\prime})^{*} (c^)/c(\widehat{c}^{\prime})^{*}/c
    \yng(3,1)\yng(3,1) \yng(3,2,1,1)\yng(3,2,1,1) \young(:1,2,3)\young(:1,2,3)  \young(:2,1,3)\young(:2,1,3)  \young(:3,1,2)\young(:3,1,2)
    \yng(2,2)\yng(2,2) \yng(3,2,2)\yng(3,2,2) {ytableau} \none \none 1
    \none \none \none
    2 3   {ytableau} \none \none 2
    \none \none \none
    1 3   {ytableau} \none \none 3
    \none \none \none
    1 2
    \yng(2,1,1)\yng(2,1,1) \yng(3,3,1)\yng(3,3,1) \young(:1,23)\young(:1,23)  \young(:2,13)\young(:2,13)
    Table 1. standard Young tableaux of skew shape (c^)/c(\widehat{c}^{\prime})^{*}/c

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