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23 24 25 26 27<meta charset="UTF-8"> 28<title>Rui Xiong</title> 29</head> 30 31 32<body> 33 34<h1>Rui Xiong</h1> 35<p style="text-align:center;" >PhD in Mathematics, my <a href="/doc/CV.pdf">CV</a>.</p> 36 37<!-- -<hr> - --> 38 39<h2>Information</h2> 40<!-- <p> --> 41 <!-- Advisor <a href="https://kirillmath.ca/">Kirill Zaynullin</a> --> 42<!-- <br> --> 43 Email 44 <!-- <a href="mailto:[email protected]">[email protected]</a> and --> 45 <a href="mailto:[email protected]">[email protected]</a> 46 (happy to chat about math!) 47</p> 48 49<p> 50 A <a href="/pic/IMG_9757.jpeg">photo</a> of me. Artists' impression of my research: <a href="/pic/PipeDream.JPG">Leisurely Pipe-Dreaming</a>. 51 </p> 52 53<p> 54My research interest is <a href="https://en.wikipedia.org/wiki/Enumerative_geometry">enumerative geometry</a> (<a href="https://en.wikipedia.org/wiki/Schubert_calculus">Schubert calculus</a>) 55and <a href="https://en.wikipedia.org/wiki/Algebraic_combinatorics">algebraic combinatorics</a>. 56I am also interested in <a href="https://en.wikipedia.org/wiki/Representation_theory">representation theory</a> and <a href="https://en.wikipedia.org/wiki/Algebraic_geometry">algebraic geometry</a>. 57</p> 58 59 <h2>Publications and Preprints</h2> 60 61 62<ol reversed> 63 64<li style = "display:none;"> 65 <b class="title">My Title</b> 66 [<a href="https://arxiv.org/abs/xxxx">arXiv</a>] 67 [<a href="/doc/xxx.pdf">talk</a>]<br> 68 <a href="https://cubicbear.github.io/">Rui Xiong</a> 69 <cite class="abstract"><br> 70 TBA 71 </cite> 72 </li> 73 74<li> 75 <b class="title">Permutation Representations on Cohomology of Toric Varieties</b> 76 [<a href="https://arxiv.org/abs/2609.06597">arXiv</a>] 77 [<a href="/doc/PermRep.pdf">talk</a>]<br> 78 <a href="https://www.researchgate.net/profile/Tao-Gui-4">Tao Gui</a>, Chushi Qin, Kaizhe Shen, 79 <a href="https://cubicbear.github.io/">Rui Xiong</a> 80 <cite class="abstract"><br> 81 Let \(G\) be a finite group acting properly by lattice automorphisms on a complete simplicial fan \(\Sigma\). An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology \(H^*(X_{\Sigma})\) of the associated toric variety \(X_{\Sigma}\) is isomorphic to a permutation representation of \(G\). We prove that Stanley's question has an affirmative answer for all smooth projective toric varieties without the properness assumption on the action. 82The proof is inspired by toric mirror symmetry. 83 </cite> 84 </li> 85 86<li> 87 <b class="title">Non-vanishing of Single, Double, and Triple Schubert Structure Constants</b> 88 [<a href="https://arxiv.org/abs/2608.17378">arXiv</a>] 89 <br> 90 Yiming Chen, 91 <a href="https://sites.google.com/view/neiljyfan/">Neil J.Y. Fan</a>, 92 <a href="https://cubicbear.github.io/">Rui Xiong</a> and 93 Ming Yao 94 <cite class="abstract"><br> 95 The Schubert vanishing problem asks whether the single Schubert coefficients \(c^w_{u,v}\) are zero. In this paper, we consider the non-vanishing problems of double Schubert coefficients \(c^w_{u,v}(t)\) and triple Schubert coefficients \(c^w_{u,v}(t;y)\). We show that the non-vanishing of \(c^w_{u,v}(t;y)\) is completely determined by the non-vanishing of single Schubert coefficients. As a byproduct, we obtain the saturation property of the triple Littlewood--Richardson coefficients \(c^\nu_{\lambda,\mu}(t;y)\). Moreover, we pose a conjecture asserting that the non-vanishing of \(c^w_{u,v}(t)\) is also determined by the non-vanishing of single or triple Schubert coefficients. We prove a one-side inclusion of the conjecture. For the reverse inclusion, we show that the conjecture holds for the following three cases: the Pieri case, the separated descents case, and the inverse Grassmannian case. 96 </cite> 97 </li> 98 99<li> 100 <b class="title">Quantized Coulomb Branches of Separated Cotangent Type and Orthosymplectic Quivers</b> 101 [<a href="https://arxiv.org/abs/2608.16091">arXiv</a>] 102 [<a href="/doc/CBSPslides.pdf">talk</a>] 103 <br> 104 <a href="https://sites.google.com/virginia.edu/yaolongshen/homepage">Yaolong Shen</a>, 105 <a href="https://changjiansu.github.io/">Changjian Su</a> and 106 <a href="https://cubicbear.github.io/">Rui Xiong</a> 107 <cite class="abstract"><br> 108 We propose a definition of the quantized Coulomb branches of separated cotangent type, and prove that the corresponding classical construction recovers the non-cotangent Coulomb branch. We also obtain a formula for quasi-minuscule monopole operators in arbitrary cotangent type. Applying these results, we compute the monopole operators for orthosymplectic quivers and construct a homomorphism from the shifted t
108wisted Yangian of split ADE type to the corresponding quantized Coulomb branch algebra. 109 </cite> 110 </li> 111 112 113<li> 114 <b class="title">A solution to Butler's positivity conjecture</b> 115 [<a href="https://arxiv.org/abs/2608.11543">arXiv</a>] 116 [<a href="/doc/ButlerConj.pdf">talk</a>]<br> 117 <a href="https://peter-guo.com"> 118 Peter L. Guo</a>, 119 <a href="https://www.kangmingyang.com"> 120 Mingyang Kang</a>, and 121 <a href="https://cubicbear.github.io/">Rui Xiong</a> 122 <cite class="abstract"><br> 123 Let \(\lambda, \mu, \nu\) be distinct partitions such that \(\lambda, \mu\subset \nu\) and \(|\nu/\lambda|=|\nu/\mu|=1\). 124We prove a longstanding conjecture on Macdonald polynomials made by Butler: the expansion of the Macdonald intersection polynomial 125\[ 126\frac{T_\lambda\widetilde{H}_\mu(X;q,t)-T_\mu\widetilde{H}_\lambda(X;q,t)} 127 {T_\lambda-T_\mu} 128\] 129in terms of the Schur function basis has coefficients in \(\mathbb{Z}_{\geq 0}[q,t]\). 130 131 </cite> 132 </li> 133 134<li> 135 <b class="title">On weight modules over truncated shifted iYangians</b> 136 [<a href="https://arxiv.org/abs/2607.27596">arXiv</a>] 137 <br> 138 <a href="https://sites.google.com/virginia.edu/yaolongshen/homepage">Yaolong Shen</a>, 139 <a href="https://math.tongji.edu.cn/info/1253/9791.htm">linliang Song</a> and 140 <a href="https://cubicbear.github.io/">Rui Xiong</a> 141 <cite class="abstract"><br> 142 Truncated shifted iYangians are a family of algebras expected to quantize certain components of affine Grassmannian islices. We introduce orientifold KLRW (oKLRW) algebras associated with quivers with involution and establish their faithful polynomial representations and diagrammatic bases. We also define KLR iYangians using double reflective KLR diagrams and construct diagrammatic realizations of the iGKLO homomorphisms. For integral parameters, we introduce interval oKLRW algebras and prove an equivalence between integral weight modules over truncated shifted iYangians and nilpotent modules over the corresponding interval oKLRW algebras. 143 </cite> 144 </li> 145 146<li> 147 <b class="title">Equivariant Schubert Calculus for Inverse Grassmannian Permutations</b> 148 [<a href="https://arxiv.org/abs/2607.16797">arXiv</a>] 149 [<a href="/doc/S-orbit-slides.pdf">talk</a>] 150 <br>Yiming Chen, 151 <a href="https://sites.google.com/view/neiljyfan/">Neil J.Y. Fan</a>, 152 <a href="https://cubicbear.github.io/">Rui Xiong</a> and 153 Ming Yao 154 <cite class="abstract"><br> 155 We give a Graham-positive expansion for the product of two double Schubert polynomials indexed by two inverse Grassmannian permutations. Surprisingly, the nonzero structure constants are double Schubert polynomials in two disjoint sets of equivariant variables. We also give a positive expansion for the product of two single Schubert polynomials indexed by a 321-avoiding permutation (e.g., a Grassmannian permutation) and an inverse Grassmannian permutation. Unexpectedly, the nonzero structure constants are Edelman--Greene coefficients. 156 </cite> 157 </li> 158 159<li> 160 <b class="title">Motivic Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells</b> 161 [<a href="https://arxiv.org/abs/2603.07269">arXiv</a>]<br> 162 <a href="https://changjiansu.github.io/">Changjian Su</a>, 163 <a href="https://cubicbear.github.io/">Rui Xiong</a> and 164 <a href="https://www.albany.edu/~cz954339/">Changlong Zhong</a> 165 <cite class="abstract"><br> 166 The open projected Richardson varieties are images of the open Richardson varieties of the complete flag variety under the canonical projection to the partial flag variety. Our main result compares the Segre motivic Chern (SMC) classes of the open projected Richardson varieties with those of the affine Schubert cells by pushing or pulling these classes to the affine Grassmannian. The main method is the recursive relation determined by the Demazure--Lusztig operators. As another application of this recursive relation, we relate the localization of the SMC classes to the twisted Kazhdan--Lusztig R-polynomials. In the case of Grassmannians, the open projected Richardson varieties are known as the open positroid varieties. We give a combinatorial formula for the SMC classes of these varieties. 167 </cite> 168 </li> 169 170<li> 171 <b class="title">Bumpless Pipe Dream Fragments -- Equivariant Geometry of Clans</b> 172 [<a href="https://arxiv.org/abs/2511.00980">arXiv</a>] 173 [<a href="/doc/K-orbit_slides.pdf">slides</a>] 174 <!-- [<a href="/doc/xxx.pdf">talk</a>]<br> --> 175 <br>Yiming Chen, 176 <a href="https://sites.google.com/view/neiljyfan/">Neil J.Y. Fan</a>, 177 <a href="https://cubicbear.github.io/">Rui Xiong</a> and 178 Ming Yao 179 <cite class="abstract"><br> 180 In this paper, we establish a new geometric setting for bumpless pipe dreams and double Schubert polynomials. Building on the notion of bumpless pipe dream fragments, we define clan polynomials as their weight generating functions. It turns out that clan polynomials arise naturally in the equivariant geometry of (\(GL_p\times GL_q\))-orbits over the flag variety \(Fl_{p+q}\) parametrized by (\(p,q\))-clans. Furthermore, we show that the coefficients in the equivariant Schubert expansion of the fundamental classes of (GLpÃGLq)-orbit closures are exactly clan polynomials, which resolves an open problem posed by Wyser and Yong. 181 </cite> 182 </li> 183 184<li> 185 <b class="title">
185Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches</b> 186 [<a href="https://arxiv.org/abs/2510.12118">arXiv</a>] 187 <br> 188 <a href="https://sites.google.com/virginia.edu/yaolongshen/homepage">Yaolong Shen</a>, 189 <a href="https://changjiansu.github.io/">Changjian Su</a> and 190 <a href="https://cubicbear.github.io/">Rui Xiong</a> 191 <cite class="abstract"><br> 192 To a quiver with involution, we study the Coulomb branch of the 3d \(\mathcal{N}=4\) involution-fixed part of the quiver gauge theory. We show that there is an algebra homomorphism from the corresponding shifted twisted Yangian to the quantized Coulomb branch algebra. This gives a new instance of 3D mirror symmetries. 193 </cite> 194 </li> 195 196<li> 197 <b class="title">Combinatorial Aspects of Elliptic Schubert Calculus</b> 198 [<a href="https://arxiv.org/abs/2510.04336">arXiv</a>] 199 [<a href="/doc/EllPDslides.pdf">talk</a>]<br> 200 <a href="https://www.albany.edu/math/faculty/cristian-lenart">Cristian Lenart</a>, 201 <a href="https://cubicbear.github.io/">Rui Xiong</a> and 202 <a href="https://www.albany.edu/~cz954339/">Changlong Zhong</a> 203 <cite class="abstract"><br> 204 The main goal of this paper is to extend two fundamental combinatorial results in Schubert calculus on flag manifolds from equivariant cohomology and K-theory to equivariant elliptic cohomology. The foundations of elliptic Schubert calculus were laid in a few relatively recent papers by Rimányi, Weber, and Kumar. They include the recursive construction of elliptic Schubert classes via generalizations of the cohomology and K-theory push-pull operators and the study of the corresponding Demazure algebra. We derive a Billey-type formula for the localization of elliptic Schubert classes (for partial flag manifolds of arbitrary type) and a pipe dream model for their polynomial representatives in the case of type A flag manifolds. The latter extends the pipe dream model for double Schubert and Grothendieck polynomials. We also study the degeneration of elliptic Schubert classes to K-theory, which recovers the corresponding classical formulas. 205 206 </cite> 207 </li> 208 209 210<li> 211 <b class="title">Quantum Schubert calculus for smooth Schubert divisors of \(\mathcal{F}\ell_n\)</b> 212 [<a href="https://arxiv.org/abs/2509.17857">arXiv</a>] 213 <br> 214 <a href="https://math.sysu.edu.cn/gagp/czli">Changzheng Li</a>, 215 Jiayu Song, 216 <a href="https://cubicbear.github.io/">Rui Xiong</a> and Mingzhi Yang 217 <cite class="abstract"><br> 218 We propose to study the quantum Schubert calculus for Schubert varieties, and investigate the smooth Schubert divisors \(X\) of the complete flag variety \(\mathcal{F}\ell_n\). We provide a Borel-type ring presentation of the quantum cohomology of \(X\). We derive the quantum Chevalley formula for \(X\) by geometric arguments. We also show that the quantum Schubert polynomials for \(X\) are the same as that for \(\mathcal{F}\ell_n\) introduced by Fomin, Gelfand and Postnikov. 219 </cite> 220 </li> 221 222 <li> 223 <b class="title">ADE diagrams, Hodge--Tate hyperplane sections and semisimple quantum cohomology</b> 224 [<a href="https://arxiv.org/abs/2509.01101">arXiv</a>] 225 [<a href="/doc/ADE_slides.pdf">talk</a>] 226 <br> 227 <a href="http://mat.puc-rio.br/~galkin/">Sergey Galkin</a>, 228 <a href="https://www.ims.cuhk.edu.hk/~leung/">Naichung Conan Leung</a>, 229 <a href="https://math.sysu.edu.cn/gagp/czli">Changzheng Li</a> 230 and 231 <a href="https://cubicbear.github.io/">Rui Xiong</a> 232 (with an appendix by <a href="https://pbelmans.ncag.info">Pieter Belmans</a> and <a href="https://math.ruhr-uni-bochum.de/fakultaet/arbeitsbereiche/algebra/research-team-reineke/team/prof-dr-markus-reineke/">Markus Reineke</a>) 233 <cite class="abstract"><br> 234 It is known that the semisimplicity of quantum cohomology implies the vanishing of off-diagonal Hodge numbers (Hodge--Tateness). We investigate which hyperplane sections of homogeneous varieties possess either of the two properties. We provide a new efficient criterion for non-semisimplicity of the small quantum cohomology ring of Fano manifolds that depends only on the Fano index and Betti numbers. We construct a bijection between Dynkin diagrams of types A, D or E, and complex Grassmannians with Hodge-Tate smooth hyperplane sections. By applying our criteria and using monodromy action, we completely characterize the semisimplicity of the small quantum cohomology of smooth hyperplane sections in the case of complex Grassmannians, and verify a conjecture of Benedetti and Perrin in the case of (co)adjoint Grassmannians. 235 </cite> 236 </li> 237 238<li> 239 <b class="title">Graham positivity of triple Schubert calculus</b> 240 [<a href="https://arxiv.org/abs/2506.09421">arXiv</a>] 241 [<a href="/doc/Triple_slides.pdf">talk</a>] <br> 242 <a href="http://faculty.bicmr.pku.edu.cn/~gaoyibo/">Yibo Gao</a> and 243 <a href="https://cubicbear.github.io/">Rui Xiong</a> 244 <cite class="abstract"><br> 245 We prove Samuel's conjecture on certain Graham positivity of the expansion coefficient of two double Schubert polynomials in three set
245s of variables by establishing a refined version of Graham's positivity theorem. As a corollary, we prove Kirillov's conjecture on the positivity of skew divided difference operators applied to Schubert polynomials. 246 </cite> 247 </li> 248 249 <li> 250 <b class="title"> 251 Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells 252 </b> 253 [<a href="https://arxiv.org/abs/2501.16172">arXiv</a>] 254 [<a href="https://doi.org/10.1093/imrn/rnaf305">journal</a>] 255 [<a href="/doc/Richardson_VS_Schubert.pdf">talk</a>] 256 [<a href="/doc/positroid_poster.pdf">poster</a>] 257 <br> 258 <i>International Mathematics Research Notices</i>, Volume 2025, Issue 19, October 2025, rnaf305<br> 259 <a href="https://math.scu.edu.cn/info/1013/4291.htm"> 260 Neil J.Y. Fan</a>, 261 <a href="https://peter-guo.com/"> 262 Peter L. Guo</a>, <a href="https://changjiansu.github.io/"> 263 Changjian Su</a> and <a href="https://cubicbear.github.io/">Rui Xiong</a> 264 265<cite class="abstract"><br> 266The open projected Richardson varieties form a stratification for the partial flag variety \(G/P\). We compare the Segre--MacPherson classes of open projected Richardson varieties with those of the corresponding affine Schubert cells by pushing or pulling these classes to the affine Grassmannian. In the case of the Grassmannian \(G/P=\operatorname{Gr}_k(\mathbb{C}^n)\), the open projected Richardson varieties are known as open positroid varieties. We obtain symmetric functions that represent the Segre--MacPherson classes of these open positroid varieties, constructed explicitly in terms of pipe dreams for affine permutations. 267</cite> 268 </li> 269 270 <li> 271 <b class="title"> 272 Hybrid Pipe Dreams for Key Polynomials 273 </b> 274 [<a href="https://arxiv.org/abs/2411.01637">arXiv</a>] 275 [<a href="https://doi.org/10.1016/j.aam.2025.102979">link</a>]<br> 276 <i>Advances in Applied Mathematics</i>, Volume 173, Part A<br> 277<!-- [<a href="/doc/xxx.pdf">talk</a>]<br> --> 278 Yihan Xiao, <a href="https://cubicbear.github.io/">Rui Xiong</a> and Haofeng Zhang 279<cite class="abstract"><br> 280We develop a family of new combinatorial models for key polynomials. It is similar to the hybrid pipe dream model for Schubert polynomials defined recently by Knutson and Udell. 281</cite> 282 283 284 <li> 285 <b class="title"> 286 Motivic Lefschetz theorem for twisted Milnor hypersurfaces 287 </b> 288 [<a href="https://arxiv.org/abs/2404.07314">arXiv</a>] 289 [<a href="/doc/Motivic_slides.pdf">talk</a>] 290 [<a href="/Compounds">code</a>]<br> 291 <a href="https://cubicbear.github.io/">Rui Xiong</a> 292 and <a href="https://kirillmath.ca/">Kirill Zaynullin</a> 293 294<cite class="abstract"><br> 295We show that the Grothendieck-Chow motive of a smooth hyperplane section \(Y\) of an inner twisted form \(X\) of a Milnor hypersurface splits as a direct sum of shifted copies of the motive of the Severi-Brauer variety of the associated cyclic algebra \(A\) and the motive of its maximal commutative subfield \(L\subset A\). The proof is based on the non-triviality of the (monodromy) Galois action on the equivariant Chow group of \(Y_L\). 296</cite> 297 </li> 298 299 <li> 300 <b class="title"> 301 A Pieri type formula for motivic Chern classes of Schubert cells in Grassmannians</b> 302 [<a href="https://arxiv.org/abs/2402.04500">arXiv</a>] 303 [<a href="/doc/Adding_Ribbons_slides.pdf">talk</a>] 304 [<a href="/doc/AddingRibbon2.pdf">talk2</a>] 305 [<a href="/doc/AddingRibbons.pdf">poster</a>] 306 [<a href="/PluckerHodge.html">code</a>]<br> 307 <a href="https://math.scu.edu.cn/info/1013/4291.htm"> 308 Neil J.Y. Fan</a>, 309 <a href="https://peter-guo.com/"> 310 Peter L. Guo</a>, <a href="https://changjiansu.github.io/"> 311 Changjian Su</a> and <a href="https://cubicbear.github.io/">Rui Xiong</a> 312 313<cite class="abstract"><br> 314We prove a Pieri formula for motivic Chern classes of Schubert cells in the equivariant K-theory of Grassmannians, which is described in terms of ribbon operators on partitions. Our approach is to transform the Schubert calculus over Grassmannians to the calculation in a certain affine Hecke algebra. As a consequence, we derive a Pieri formula for Segre motivic classes of Schubert cells in Grassmannians. We apply the Pieri formulas to establish a relation between motivic Chern classes and Segre motivic classes, extending a well-known relation between the classes of structure sheaves and ideal sheaves. As another application, we find a symmetric power series representative for the class of the dualizing sheaf of a Schubert variety. 315</cite> 316 </li> 317 318 <li> 319 <b class="title"> 320 On the Formal Peterson subalgebra and its dual</b> 321 [<a href="https://arxiv.org/abs/2312.03965">arXiv</a>] 322 [<a href="https://doi.org/10.4153/S0008414X25101259">journal</a>]<br> 323<i>Canadian Journal of Mathematics</i>, accepted<br> 324 <a href="https://cubicbear.github.io/">Rui Xiong</a> 325, <a href="https://www.albany.edu/~cz954339/">Changlong Zhong</a> and <a href="https://kirillmath.ca/">Kirill Zaynullin</a> 326 327<cite class="abstract"><br> 328In the present notes, we study a generalization of the Peterson subalgebra to an oriented (generalized) cohomology theory which we call the formal Peterson subalgebra. Observe that by recent results of Zhong the dual of the formal Peterson algebra provides an algebraic model for the oriented cohomology of the affine Grassmannian.<br> 329Our first result shows that the centre of the formal affine Demazure algebra generates the formal Peterson subalgebra. Our second observation is motivated by the Peterson conjecture. We show that a certain localization of the formal Peterson subalgebra for the extended Dynkin diagram of type \(A^1\) provides an algebraic model for 'quantum' oriented cohomology of the projective line. Our last result can be viewed as an extension of the previous results on Hopf algebroids of structure algebras of moment graphs to the case of affine root systems. We prove that the dual of the formal Peterson subalgebra (an oriented cohomology of the affine Grassmannian) is the \(0\)th Hochshild homology of the formal affine Demazure algebra. 330</cite> 331 </li> 332 333 <li> 334 <b class="title"> 335 Bumpless pipe dreams meet puzzles</b> 336 [<a href="https://arxiv.org/abs/2309.00467" >arXiv</a>] 337 [<a href="/doc/PipiPuzzle.pdf">poster</a>] 338 [<a href="/doc/puzzles_slides.pdf">talk</a>] 339 [<a href="PipePuzzle.html">code</a>] 340 [<a href="https://doi.org/10.1016/j.aim.2025.110113">journal</a>] 341 <br> 342<i>Advances in Mathematics</i>, Volume 463, March 2025, 110113.<br> 343 <a href="https://math.scu.edu.cn/info/1013/4291.htm">Neil J.Y. Fan</a>, <a href="https://peter-guo.com/">Peter L. Guo</a> and 344 <a href="https://cubicbear.github.io/">Rui Xiong</a> 345 346<cite class="abstract"><br> 347Knutson and Zinn-Justin recently found a puzzle rule for the expansion of the product \(\mathfrak{G}_u(x,t)\cdot \mathfrak{G}
347_v(x,t)\) of two double Grothendieck polynomials indexed by permutations with separated descents. We establish its triple Schubert calculus version in the sense of Knutson and Tao, namely, a formula for expanding \(\mathfrak{G}_u(x,y)\cdot \mathfrak{G}_v(x,t)\) in different secondary variables. Our rule is formulated in terms of pipe puzzles, incorporating both the structures of bumpless pipe dreams and classical puzzles. As direct applications, we recover the separated-descent puzzle formula by Knutson and Zinn-Justin (by setting \(y=t\) and the bumpless pipe dream model of double Grothendieck polynomials by Weigandt (by setting \(v=\operatorname{id}\) and \(x=t\)). Moreover, we utilize the formula to partially confirm a positivity conjecture of Kirillov about applying a skew operator to a Schubert polynomial. 348</cite> 349 350 </li> 351 352 <li> 353 <b class="title"> 354 Automorphisms of the Quantum Cohomology of the Springer Resolution and Applications</b> 355 [<a href="https://arxiv.org/abs/2304.07173">arXiv</a>] 356 [<a href="/doc/CotQum.pdf">poster</a>] 357 [<a href="/doc/CMS_QHTFDL.pdf">talk</a>] 358 [<a href="https://doi.org/10.1016/j.aim.2024.109577">journal</a>]<br> 359<i>Advances in Mathematics</i>, Volume 442, April 2024, 109577.<br> 360 <a href="https://changjiansu.github.io/"> 361 Changjian Su</a>, <a href="https://math.sysu.edu.cn/gagp/czli"> 362 Changzheng Li</a> and <a href="https://cubicbear.github.io/">Rui Xiong</a> 363 364 365<cite class="abstract"><br> 366In this paper, we introduce quantum Demazure--Lusztig operators acting by ring automorphisms on the equivariant quantum cohomology of the Springer resolution. Our main application is a presentation of the torus-equivariant quantum cohomology in terms of generators and relations. We provide explicit descriptions for the classical types. We also recover Kim's earlier results for the complete flag varieties by taking the Toda limit. 367</cite> 368 </li> 369 370 371 <li><b class="title"> 372 Structure algebras, Hopf algebroids and oriented cohomology of a group</b> 373 [<a href="https://arxiv.org/abs/2303.02409">arXiv</a>] 374 <span style = "display:none;"> [<a href="/doc/coprod.pdf">poster</a>]</span> 375 [<a href="/doc/CMS_coprod.pdf">talk</a>]<br> 376<i>Mathematical Research Letter</i>, accepted.<br> 377 <!--authors--> 378 <a href="https://sites.google.com/site/martinalanini5/home"> 379 Martina Lanini</a>, <a href="https://cubicbear.github.io/">Rui Xiong</a> and 380 <a href="https://kirillmath.ca/"> 381 Kirill Zaynullin</a> 382 383<cite class="abstract"><br> 384We prove that the structure algebra of a Bruhat moment graph of a finite real root system is a Hopf algebroid with respect to the Hecke and the Weyl actions. We introduce new techniques (reconstruction and push-forward formula of a product, twisted coproduct, double quotients of bimodules) and apply them together with our main result to linear algebraic groups, to generalized Schubert calculus, to combinatorics of Coxeter groups and finite real root systems. As for groups, it implies that the natural Hopf-algebra structure on the algebraic oriented cohomology \(h(G)\) of Levine-Morel of a split semi-simple linear algebraic group G can be lifted to a 'bi-Hopf' structure on the T-equivariant algebraic oriented cohomology of the complete flag variety. As for the Schubert calculus, we prove several new identities involving (double) generalized equivariant Schubert classes. As for finite real root systems, we compute the Hopf-algebra structure of 'virtual cohomology' of dihedral groups \(I_2(p)\), where \(p\) is an odd prime. 385</cite> 386 </li> 387 388 <li><b class="title"> 389 Pieri and MurnaghanâNakayama type Rules for Chern classes of Schubert Cells</b> 390 [<a href="https://arxiv.org/abs/2211.06802">arXiv</a>] 391 [<a href="/doc/CSMdomino.pdf">poster</a>] 392 [<a href="/doc/Domino.pdf">talk</a>] 393 [<a href="https://doi.org/10.1007/s00029-025-01112-y">journal</a>]<br> 394 <!--authors--> 395 <i>Selecta Math. (N.S.)</i> Volume 32 (2026)<br> 396 <a href="https://math.scu.edu.cn/info/1013/4291.htm"> 397 Neil J.Y. Fan</a>, 398 <a href="https://peter-guo.com/"> 399 Peter L. Guo</a> and <a href="https://cubicbear.github.io/">Rui Xiong</a> 400 401 <cite class="abstract"><br> 402We develop Pieri type as well as Murnaghan--Nakayama type formulas for equivariant Chern--Schwartz--MacPherson classes of Schubert cells in the classical flag variety. These formulas include as special cases many previously known multiplication formulas for Chern--Schwartz--MacPherson classes or Schubert classes. We apply the equivariant Murnaghan--Nakayama formula to the enumeration of rim hook tableaux. 403</cite> 404 </li> 405 406 <li><b class="title"> 407 Equivariant log-concavity and equivariant Kähler packages</b> 408 [<a href="https://arxiv.org/abs/2205.05420">arXiv</a>] 409 [<a href="https://doi.org/10.1016/j.jalgebra.2024.05.018">journal</a>]<br> 410 <i>Journal of Algebra</i>, Volume 657, November 2024, 379-401.<br><a href="https://www.researchgate.net/profile/Tao-Gui-4">Tao Gui</a> and <a href="https://cubicbear.github.io/">Rui Xiong</a> 411<cite class="abstract"><br> 412We show that the exterior algebra \(\Lambda_{\mathbb{R}}[a_1,\cdots,a_n]\), which is the cohomology of the torus \(T=(S^1)^n\), and the polynomial ring \(\mathbb{R}
412[t_1,\cdots,t_n]\), which is the cohomology of the classifying space \(B(S^1)^n=(\mathbb{C}P^\infty)^n\), are \(S_n\)-equivariantly log-concave. We do so by explicitly giving the \(S_n\)-representation maps on the appropriate sequences of tensor products of polynomials or exterior powers and proving that these maps satisfy the hard Lefschetz theorem. Furthermore, we prove that the whole Kähler package, including algebraic analogies of the Poincaré duality, hard Lefschetz, and Hodge-Riemann bilinear relations, holds on the corresponding sequences in an equivariant setting. 413</cite> 414 </li> 415 416 417 <li> 418 <b class="title"><abbr title="Introduction to set theory, topology and algebra">éåè®ºãææä¸ä»£æ°åæ¥</abbr></b> (textbook) 419 [<a href="/doc/MathABC(20190823).pdf">draft</a>] 420 <br><i>Tsinghua University Press. ISBNï¼9787302541646 </i> 421 <br><a href="https://faculty.sdu.edu.cn/liushoumin/en/index/644416/list/index.htm">Shoumin Liu</a> and Rui Xiong 422 </li> 423</ol> 424 425<h2>Seminars and Notes</h2> 426<ul> 427 <li><a href="/doc/top_trans.pdf" title="Topology in Four Days">ææåæ¥è°</a> (translated from <a href="https://link.springer.com/chapter/10.1007/978-94-010-0446-6_3">Topology in Four Days</a>)</li> 428 <li style = "display:none;"><a href="/doc/TheFinestOfGeometry(20191124).pdf" title="The finest of geometry">å ä½å¦è§æ¢</a> (æ©æç¬è®°, æä¸æ´æ°)</li> 429 <li><a href="/doc/SpeSeq.pdf">Spectral Sequences, My Homlogical Saw</a></li> 430 <li><a href="/doc/Linear_Algebra_Problems.pdf">Problems and Topics in Linear Algebra</a></li> 431 <li style = "display:none;"><a href="/doc/QuiverNote.pdf">Note on Quiver Representations</a> (2020)</li> 432 <li style = "display:none;"><a href="/doc/BorelWeil.pdf">Borel--Weil theorem and applications</a> (2020)</li> 433 <li style = "display:none;"><a href="/qcoh.html">Quantum cohomology for combinatorists</a> (2023)</li> 434 435<!-- <li>(2021) Spectral Sequences, My Homlogical Saw 436<!-- [<a href="/SpeSeq">page</a>] --> 437<!-- [<a href="/doc/SpeSeq.pdf">notes</a>]</li> --> 438 <li>(2022) Toric Varieties and their Applications 439 [<a href="/Toric.html">page</a>] 440 [<a href="/doc/Toric.pdf">notes</a>] </li> 441 <li>(2023) Geometric Representation Theory Seminar 442 [<a href="/GRT.html#2023Fall">page</a>]</li> 443 <li>(2024) Affine Weyl Groups 444 [<a href="/affine.html">page</a>] 445 [<a href="/doc/affineNotes.pdf">notes</a>]</li> 446 <li>(2024) Geometric Representation Theory Seminar 447 [<a href="/GRT.html#2024Fall">page</a>]</li> 448 <li>(2025) Macdonald polynomials 449 [<a href="http://faculty.bicmr.pku.edu.cn/~gaoyibo/learning.html">page</a>] 450 [<a href="/doc/Polynomials.pdf">notes</a>]</li> 451 <li>(2025) Geometric Representation Theory Seminar 452 [<a href="/GRT.html#2025Fall">page</a>]</li> 453 <li>(2025) Richardson Varieties 454 [<a href="http://faculty.bicmr.pku.edu.cn/~gaoyibo/learning.html">page</a>] 455 </li> 456 <li>(2026) Solvable lattice models in combinatorics 457 [<a href="https://gtbarkley.org/seminars/latticemodels26/">page</a>] 458 </li> 459 <li>(2026) Geometric Representation Theory Seminar 460 [<a href="https://yaolong-shen-mathematics.yaolongshen3555.chatgpt.site/seminar#2026Fall">page</a>]</li> 461 462</ul> 463 464<h2>Other</h2> 465 466<ul> 467 <li> 468<a href="https://github.com/CubicBear/TooYoung">TooYoung</a> <br> 469 A small package of LaTeX to draw Young diagrams.</li> 470 471 <li> 472<a href="/code/HPD">Pipe Dream Generator (ordinary/bumpless/hybrid) </a> <br> 473 An online generator of 474 <a href="https://math.mit.edu/~rstan/pubs/pubfiles/94.pdf">ordinary</a>, 475 <a href="http://www.arxiv.org/abs/1806.11233">bumpless</a> and 476 <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2023/89.pdf">hybrid</a> pipe dreams. 477</li> 478 <li> 479<a href="/Sage.html">Tutorial on SageMath</a><br> 480 Sample codes of SageMath. 481</li> 482 483 <li style = "display:none;">Tutorial on LaTeX<br> 484 Thinking.</li> 485 486</ul> 487 488
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