Presentations of Grothendieck constructions

A presentation of the symmetric Grothendieck-Witt group of local rings over $\mathbb{F}_2$

Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring $\mathrm{GW}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field $\mathbb{F}_2$. This completes a recent work by Rogers and Schlichting, where an explicit presentation of $\mathrm{GW}^{\mathrm{s}}(R)$ is given when the residue field is different from $\mathbb{F}_2$. We then use this result to compute the symmetric Grothendieck-Witt rings for the sequences of local rings $\mathbb{Z}/2^n\mathbb{Z}$ and $\mathbb{F}_2[x]/(x^n)$.

关于局部环上对称双线性型的 Grothendieck-Witt 群的展示

On the presentation of the Grothendieck-Witt group of symmetric bilinear forms over local rings

本文证明了剩余域不为F2的交换局部环上内积空间的链引理，并借此确立了该环上对称双线性型Grothendieck-Witt群的通常展示成立。

Free-fermions and skew stable Grothendieck polynomials

Skew stable Grothendieck polynomials are $K$-theoretic analogues of skew Schur polynomials. We give a free-fermionic presentation of skew stable Grothendieck polynomials and their dual symmetric functions. By using our presentation, we derive a family of determinantal formulas, which are $K$-analogues of the Jacobi-Trudi formula for skew Schur functions. We also introduce a combinatorial method to calculate certain expansions of skew (dual) stable Grothendieck polynomials by using the non-commutative supersymmetric Schur functions.

自由费米子与典范格罗滕迪克多项式

Free-fermions and canonical Grothendieck polynomials

本文用自由费米子表示精细典范格罗滕迪克多项式及其斜版本，并由此推导出斜柯西恒等式、分支规则、展开公式与积分公式等多项恒等式。